IFT-UAM/CSIC-13-065
Discrete Flavor Symmetries in D-brane models
arXiv:1306.1284v2 [hep-th] 25 Jun 2013
Fernando Marchesano,1 Diego Regalado1,2 and Liliana Vázquez-Mercado3
1
2
Instituto de Fı́sica Teórica UAM/CSIC, Cantoblanco, 28049 Madrid, Spain
Departamento de Fı́sica Teórica, Universidad Autónoma de Madrid, 28049 Madrid, Spain
3
Departamento de Fı́sica, DCI, Campus León, Universidad de Guanajuato,
C.P. 37150 Guanajuato, México.
Abstract
We study the presence of discrete flavor symmetries in D-brane models of particle
physics. By analyzing the compact extra dimensions of these models one can determine when such symmetries exist both in the context of intersecting and magnetized
D-brane constructions. Our approach allows to distinguish between approximate
and exact discrete symmetries, and it can be applied to compactification manifolds
with continuous isometries or to manifolds that only contain discrete isometries,
like Calabi-Yau three-folds. We analyze in detail the class of rigid D-branes models
based on a Z2 ×Z02 toroidal orientifold, for which the flavor symmetry group is either
the dihedral group D4 or tensor products of it. We construct explicit Pati-Salam
examples in which families transform in non-Abelian representations of the flavor
symmetry group, constraining Yukawa couplings beyond the effect of massive U(1)
D-brane symmetries.
Contents
1 Introduction
1
2 Intersecting branes and the Z2 × Z02 orbifold
5
2.1
Branes at angles in the Z2 × Z02 orbifold . . . . . . . . . . . . . . . . . . .
5
2.2
Chirality and the orbifold projection . . . . . . . . . . . . . . . . . . . . .
8
3 Discrete flavor symmetries for intersecting branes
11
3.1
Flavor symmetries on the torus . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2
Flavor symmetries on Z2 × Z02 . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3
Exact versus approximate symmetries . . . . . . . . . . . . . . . . . . . . . 20
4 Discrete flavor symmetries for magnetized branes
22
4.1
Non-Abelian flavor symmetries from magnetization . . . . . . . . . . . . . 23
4.2
Wavefunction representations in T6 /Z2 × Z2 . . . . . . . . . . . . . . . . . 28
5 Examples
32
5.1
Orientifolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2
A global Pati-Salam four-generation model . . . . . . . . . . . . . . . . . . 35
5.3
A local Pati-Salam three-generation model . . . . . . . . . . . . . . . . . . 42
6 Conclusions and outlook
44
A The Z2 × Z02 orbifold
47
B Flavor symmetries from dimensional reduction
51
C Abelian discrete gauge symmetries in Z2 × Z02
55
1
Introduction
Discrete flavor symmetries are often invoked in the particle physics literature in order
to explain different patterns of quark and lepton masses and mixings, as well as to constrain the flavor structure of supersymmetric extensions of the Standard Model. In a
1
purely field theory approach, one may simply consider the whole set of possible discrete
family-dependent symmetries compatible with the Standard Model or its 4d field theory extensions, and then analyze in detail those leading to interesting physics [1]. One
may then try to obtain a more geometric understanding of such symmetries via higherdimensional field theories (see e.g. [2]), identifying the flavor symmetries with the discrete
symmetries of the compactified extra dimensions.
Alternatively, one may look for the presence of discrete flavor symmetries in particle
physics models based on string theory, aiming for a microscopic description of their origin.
For the case of heterotic orbifolds this has been addressed in [3], obtaining a classification
of possible flavor symmetries in this particular context. In general, one expects that
the more restrictive framework of string theory will select a limited number of flavor
symmetry groups that are compatible with a realistic particle physics model, as well as
specific representations for its matter fields.
Exploring this more restricted scenario is however not the only motivation to realize
discrete flavor symmetries in string theory. Given a 4d particle physics model embedded
into string theory, one should be able to determine if a flavor symmetry is exact or
approximate. If a symmetry is exact, and because string theory includes quantum gravity,
there are strong arguments indicating that it must be realized as a 4d gauge symmetry
(see e.g. [4,5] and references therein). If on the contrary a symmetry is only approximate,
string theory should provide a well-defined answer for the scale at which this symmetry
is broken, by which mechanism, and how does the breaking affect the couplings of the 4d
effective theory.
The study of discrete gauge symmetries in string theory has been recently undertaken
in a series of papers [6–10], in which the basic strategy has been to embed the discrete
gauge symmetry in a continuous one that is typically broken at the string scale.1 From all
the stringy setups that have been analyzed in these references a particularly interesting one
in terms of flavor is given by models of magnetized D-branes in toroidal compactifications.
There the above strategy allows to derive a 4d effective Lagrangian with a manifest discrete
flavor gauge symmetry [8], and the continuous group in which the discrete symmetry is
embedded involves the continuos isometries of the toroidal background.
1
See [11–13] for applications to specific string theory models.
2
The purpose of this paper is to study from a more general perspective the presence
of discrete flavor symmetries in D-brane models, using a more direct criterion that does
not involve embedding the discrete flavor symmetry in any continuous group. The reason
to do so is that generic D-brane vacua involve compactification manifolds without any
continuous isometry (like Calabi-Yau threefolds) and there the analysis made in [8] does
not apply. Now, while continuous isometries are absent in them, Calabi-Yau manifolds
do possess discrete isometries, which will translate into discrete gauge symmetries of the
4d effective action. In addition, in manifolds with non-trivial torsional one-cycles one will
obtain 4d discrete gauge symmetries arising from discrete transformations of the NSNS
B-field.2 As we will now argue, the flavor symmetry group of a D-brane model can be
understood in terms of a subgroup of this Calabi-Yau discrete gauge symmetry group.
When building a D-brane model one considers D(p + 3)-branes filling up 4d space-time
and wrapping specific p-cycles of the Calabi-Yau manifold M. Then, while the action
of the isometry group leaves M invariant, it may not leave invariant the set of p-cycles
that the D-branes wrap. In that case the Calabi-Yau gauge symmetry group generated
by isometries will be broken down to the subgroup that also leaves invariant the Dbrane content of the model. A similar statement can be made for the gauge symmetries
generated by B-field transformations. In general, given a D-brane model in a Calabi-Yau, a
4d discrete symmetry group will arise from those transformations of the metric and B-field
that leave invariant both the closed and open string backgrounds of the compactification.
The open string zero modes stretched between the background D-branes will transform
non-trivially under transformations of the B-field and metric, and so this discrete gauge
symmetry will act as a flavor symmetry group in the 4d effective theory. Finally, from
the results of [8] one can see that in the presence of D-branes the action of the generators
of discrete isometries and B-field transformations do not commute, and so one typically
ends up with a non-Abelian discrete flavor symmetry group generated by these bulk
transformations.
2
Moreover, in type II string theory models discrete gauge symmetries are obtained from reducing a RR
(p + 1)-form on a Calabi-Yau manifold M with torsional p-cycles, that is such that Tor Hp (M, Z) 6= 0 [6].
In the following we will ignore these RR discrete gauge symmetries, since typically only very massive
states unrelated to the Standard Model fields are charged under them.
3
While it is non-trivial to construct explicit Calabi-Yau models, one can illustrate all the
above statements by means of D-brane models on toroidal orbifold backgrounds. Toroidal
orbifolds are very simple examples of compactification manifolds with discrete isometries,
which are realised as certain permutations of the orbifold fixed points. Still, they have been
shown to be a fruitful framework to construct semi-realistic D-brane models of particle
physics, and specially by considering models of intersecting D6-branes [14, 15].
For concreteness we will focus our discussion in a particular orbifold background,
namely the type IIA Z2 × Z02 orientifold with rigid intersecting D6-branes analyzed in [16].
While quite simple, this class of models allows to construct several semi-realistic examples
with non-trivial flavor symmetries based on the dihedral group D4 , as we show explicitly.
We also find that, given a set of D6-branes it is straightforward to detect the presence of
a discrete flavor symmetry for them in terms of their pairwise intersection numbers. This
simple description allows in turn to characterize a useful notion of exact and approximate
discrete symmetry, as we briefly discuss and illustrate via explicit examples.
An interesting feature of this background is that it has a simple dual description in
terms of a type I orbifold with internal magnetic fluxes [17, 18], to which our approach to
detect flavor symmetries can also be applied. This T-dual description allows to make direct
contact with the results of [19–23]. In [21] the presence of discrete flavor symmetries were
detected by analyzing the zero mode wavefunctions of magnetized D-brane models [24].
In our approach wavefunctions are not necessary to detect the flavor symmetry, but they
are still important to compute the transformation properties of matter fields. We perform
a general analysis of the possible family representations valid for both intersecting and
magnetized D-brane models, obtaining agreement with previous results.
This paper is organized as follows. In section 2 we review the basic features of the
D6-brane models of [16], and reproduce the chiral index between two D6-branes with a
different approach. In section 3 we discuss the appearance of discrete flavor symmetries
in this background using a simple geometric description, which allows to characterize the
notion of approximate flavor symmetry. In section 4 we turn to the dual framework of
magnetized D-brane models, reproducing the same flavor symmetry group and classifying
the different family representations. We illustrate all the above results via a couple of
Pati-Salam examples in section 5, and leave our conclusions for section 6.
4
Several technical details have been relegated to the appendices. Appendix A contains
more technical results of [16] on the Z2 × Z02 orientifold background. Appendix B derives
the effective Lagrangian describing the discrete flavor symmetries for intersecting and
magnetized D-branes in T2n . Finally, Appendix C describes the computation of Abelian
flavor-independent discrete gauge symmetries in Z2 × Z02 .
2
Intersecting branes and the Z2 × Z02 orbifold
Our examples of D-brane models with discrete flavor symmetries will be based on the
toroidal Z2 × Z02 orientifold background analyzed in [16] (see also [17, 18, 25]). As pointed
out in there, in this background one can reproduce the main features of realistic D-brane
models in Calabi-Yau compactifications, obtaining N = 1 chiral vacua made up of rigid
D-branes. In the following we will briefly review the construction of this class of models,
emphasizing those features which are more relevant for the analysis of discrete flavor
symmetries. In doing so we will follow the notation and formalism of [16], which was
mainly developed for models of intersecting D6-branes. Nevertheless, such models have a
well-known T-dual description in terms of magnetized D-branes, a fact that we will exploit
in section 4 to understand discrete flavor symmetries from an alternative viewpoint closer
to the analysis of [8].
2.1
Branes at angles in the Z2 × Z02 orbifold
As emphasized in the literature (see [14, 15] for reviews on the subject) a quite successful
approach to construct particle physics models from string theory is by considering models
of intersecting D6-branes in type IIA Calabi-Yau compactifications. The simplest setup in
which this approach can be implemented is by taking the compactification space to be a
factorized six-torus T6 = (T2 )1 ×(T2 )2 ×(T2 )3 and adding sets of D6-branes that fill up 4d
Minkowski space and wrap different three-dimensional slices of these six extra dimensions.
Typically one considers that each D6-brane wraps a product of three one-cycles on the
factorized T6 , namely
[Πα ] =
3
O
niα [ai ] + miα [bi ]
i=1
5
niα , miα ∈ Z and coprime
(2.1)
where [ai ], [bi ] correspond to the two fundamental one-cycles of (T2 )i . We show in figure
1.i) an example of two of these D6-branes a and b wrapping the three-cycles Πa and Πb
respectively. If we now wrap Na D6-branes on top of the three-cycle Πa and Nb on top
of Πb we will have a 4d U (Na ) × U (Nb ) gauge group upon dimensional reduction, with
a 4d chiral fermion in the (Na , N̄b ) representation at each intersection point. Hence, by
considering several sets of D6-branes one can construct 4d effective theories similar to the
Standard Model or extensions thereof [26–30].
y
y3
y2
y1
x1
1
y
x1
2
i)
x2
ii)
x2
x3
y
3
x3
Figure 1: Branes with wrapping numbers (nia , mia ) = (1, 1) ⊗ (1, −1) ⊗ (2, −1) (red) and
(nib , mib ) = (−1, −3) ⊗ (−1, −1) ⊗ (2, 1) (green) with i) generic values of the position
moduli and ii) stuck at the fixed points in the Z2 × Z02 orbifold. The number of chiral
6
T
families for i) is Iab
= (−2) × (−2) × 4 = 16 and for ii) is Iab = 4.
In this setup the relative orientation of a pair of D6-branes is specified in terms of
i
three angles θab
, one per two-torus (T2 )i . One can render these two D6-branes mutually
BPS by applying certain conditions to these angles [31]. However, in order to construct a
consistent four-dimensional chiral and N = 1 supersymmetric (and hence stable) model
one needs the presence of negative tension objects like O6-planes and to replace the
compactification manifold T6 by a toroidal orbifold of the form T6 /Γ, with Γ a discrete
symmetry group [32]. We will leave the effects of adding the O6-planes for section 5 and
6
focus here on the implications of having D6-branes at angles in a toroidal orbifold rather
than on T6 .
In particular we will consider type IIA string theory compactified on the background
T6 /Z2 × Z02 , the Z2 generators acting as


 z1 → −z1
Θ:
z2 → −z2

 z →z
3
3


 z1 → z1
0
Θ :
z2 → −z2

 z → −z
3
3
(2.2)
on the three complex coordinates of T6 = (T2 )1 × (T2 )2 × (T2 )3 . This is the sort of
background considered in [16] in order to construct semi-realistic models made of rigid
D6-branes. These rigid or fractional D6-branes wrap three-cycles that are left invariant
by (2.2) and so, unlike in the case of T6 , it is not possible to displace them transversely.3
As illustrated in figure 1.ii) on each (T2 )i a fractional D6-brane goes through two fixed
points of the action zi → −zi . One can determine which are these two fixed points in
terms of the wrapping numbers (nia , mia ), as indicated in table 1. From the effective field
theory viewpoint the fact that a D6-brane a is rigid implies that at low energies there will
be no multiplets in the adjoint representation of U (Na ). One can then build chiral N = 1
models where non-Abelian gauge groups are asymptotically free [16], a required feature
for realistic models in Calabi-Yau compactifications.
(ni , mi )
Fixed points on (T2 )i
(odd, odd)
{1, 4} or {2, 3}
(odd, even)
{1, 3} or {2, 4}
(even, odd)
{1, 2} or {3, 4}
Table 1: Fixed points of a 1-cycle on a T2 /Z2 in terms of its wrapping numbers.
3
The fact that fractional D6-branes are fully rigid is a consequence of the choice of discrete torsion in
the Z2 × Z02 orbifold. Such choice implies that this orbifold contains collapsed three-cycles at the fixed
loci of (2.2), see appendix A and ref. [16] for more details. In a Z2 × Z2 orbifold with the opposite choice
of discrete torsion fractional D6-branes are not rigid [32].
7
2.2
Chirality and the orbifold projection
Let us now consider Na D6-branes wrapping the three-cycle Πa of T6 and Nb of them
wrapping Πb , with wrapping numbers
Πa : (n1a , m1a ) (n2a , m2a ) (n3a , m3a )
(2.3)
Πb : (n1b , m1b ) (n2b , m2b ) (n3b , m3b )
as stated above, this D6-brane sector yields a 4d U (Na ) × U (Nb ) gauge group at low
energies, together with N = 1 chiral multiplets in the (Na , N̄b ) representation, one per
each point of intersection of these two three-cycles. The chirality and multiplicity of these
multiplets is given respectively by the sign and by the absolute value of the topological
intersection number Iab , which in the case of T6 = (T2 )1 × (T2 )2 × (T2 )3 is given by
T6
Iab
=
1 2 3
Iab
Iab Iab
3
Y
=
(nia mib − nib mia )
(2.4)
i=1
If we now consider rigid D6-branes in T6 /Z2 × Z02 the chiral spectrum will be different,
because the Z2 × Z02 action relates several of the intersection points of the two D6-branes
and these will no longer be independent degrees of freedom. In particular, one should
project out all those zero modes that are not invariant under the Z2 × Z02 orbifold action,
as we now describe.
6
T
Let us consider the case where Iab
6= 0, so that in the toroidal case we have a net
number of chiral fermions in the ab sector. Since the D6-branes wrap BPS and factorizable
6
T
three-cycles, the massless spectrum in the ab sector is given by |Iab
| 4d chiral fermions in
6
T
the representation (Na , N̄b ), whose 4d chirality is given by sign(Iab
). In fact, by applying
the usual CFT rules for computing the open string spectrum between two intersecting
D-branes [28, 31], one can associate to each intersection the piece of 10d massless fermion
whose SO(8) weight representation is given by [33]
rab = (r0 ; r1 , r2 , r3 ) =
1
(s1 s2 s3 ; −s1 , −s2 , −s3 )
2
(2.5)
where the first entry indicates the 4d chirality, and the other three correspond to the
compact extra dimensions. Here si = sign(ϑiab ), with ϑiab the angle of intersection in (T2 )i
i
measured in anti-clockwise sense. Notice that si = sign(Iab
) and so 4d chirality is indeed
6
6
T
T
given by sign(Iab
). In our conventions Iab
> 0 corresponds to 4d left-handed fermions.
8
When introducing the orbifold projection, some of these massless fields will be projected out. In particular, we must require that the internal fermionic wavefunctions are
invariant under the action of the Z2 × Z02 generators. These act on a fermion with Lorentz
indices (2.5) as
Θ : Ψ(z1 , z2 , z3 ) 7→ eiπ(r1 −r2 ) Ψ(−z1 , −z2 , z3 ) = s1 s2 Ψ(−z1 , −z2 , z3 )
Θ0 : Ψ(z1 , z2 , z3 ) 7→ eiπ(r2 −r3 ) Ψ(−z1 , −z2 , z3 ) = s2 s3 Ψ(z1 , −z2 , −z3 )
(2.6)
A generic open string wavefunction, which are basically delta functions localized at the
intersection points, will not be invariant under such transformations, and so one must
form linear combinations that transform appropriately under internal coordinate reversal.
y
y
1
7
6
2
z → −z
5
3
4
4
3
5
2
6
1
7
0
0
x
x
Figure 2: Space inversion in one of the tori and D-branes with wrapping numbers (1, 3) and
(3, 1). Some of the intersection points are invariant while others get exchanged. The even
combination of points are {0, 1 + 7, 2 + 6, 5 + 3, 4} and the odd ones {1 − 7, 2 − 6, 5 − 3}.
Notice that the number of even and odd points is in agreement with (2.7).
In order to describe these combination of wavefunctions let us first consider two Dbranes wrapping 1-cycles of (T2 )i going through the origin, and see how their intersection
points transform under the Z2 action generated by z 7→ −z. As shown in figure 2,
one can take linear combinations of delta functions at the intersection points, in order
to form wavefunctions which are even and odd under the orbifold action. Such linear
combinations have coefficients ±1, and the number of even and odd points for a given
i
intersection number Iab
is given by
Iei =
1 i
Iab + si ρi
2
Ioi =
9
1 i
Iab − si ρi
2
(2.7)
i
where si = sign(Iab
) and
(
ρi ≡
i
odd
1 for Iab
i
2 for Iab even
(2.8)
Going back to intersecting D6-brane on (T2 )1 × (T2 )2 × (T2 )3 /Z2 × Z02 , from (2.6) it
is clear that we need to impose that our wavefunctions satisfy
Ψ(z1 , z2 , z3 ) = s1 s2 Ψ(−z1 , −z2 , z3 ) = s2 s3 Ψ(z1 , −z2 , −z3 )
(2.9)
i
we will have to
and so depending on the signs of the two-tori intersection numbers Iab
B
6= 0 and
impose different projections. In particular we have that for Iab
- s1 s2 > 0, s2 s3 > 0
=⇒
Ψ = ψej1 ψej2 ψej3 or ψoj1 ψoj2 ψoj3
- s1 s2 > 0, s2 s3 < 0
=⇒
Ψ = ψej1 ψej2 ψoj3 or ψoj1 ψoj2 ψej3
- s1 s2 < 0, s2 s3 > 0
=⇒
Ψ = ψoj1 ψej2 ψej3 or ψej1 ψoj2 ψoj3
- s1 s2 < 0, s2 s3 < 0
=⇒
Ψ = ψej1 ψoj2 ψej3 or ψoj1 ψej2 ψoj3
where ψeji runs over even combinations of intersection points on (T2 )i , and ψoji is an odd
combination of delta-wavefunctions in (T2 )i . For the first case above we have that the
number of generations after the orbifold projection is given by Iab = Ie1 Ie2 Ie3 + Io1 Io2 Io3 or
1 1 2 3
1
2
3
Iab Iab Iab + s2 s3 ρ2 ρ3 Iab
+ s1 s3 ρ1 ρ3 Iab
+ s1 s2 ρ1 ρ2 Iab
4
Iab =
Hence, after imposing that s1 s2 > 0 and s2 s3 > 0 we recover the result
Iab =
1 1 2 3
1
2
3
Iab Iab Iab + Iab
ρ2 ρ3 + Iab
ρ1 ρ3 + Iab
ρ1 ρ2
4
(2.10)
A different choice of signs s1 , s2 , s3 will select different parities for the wavefunctions of
each two-torus, and so a different total number of chiral fermions. Nevertheless, the final
expression for Iab will again be given by (2.10). Notice that this result matches eq.(A.10),
which has been obtained in appendix A by means of the topological techniques of [16].
The same statement holds if we consider the case where the toroidal intersection
6
T
number Iab
vanishes, as we now briefly discuss. Let us for instance consider the case
1
2 3
where only Iab
= 0.4 Then instead of (2.5) we have Iab
Iab fermions of the form
rab =
4
1
(s2 s3 ; −, −s2 , −s3 )
2
and
1
(−s2 s3 ; +, −s2 , −s3 )
2
(2.11)
i
The case with two vanishing intersection numbers Iab
does not correspond to D6-branes preserving
i
N = 1 supersymmetry, and will not be considered here, while the case which all three Iab
= 0 is trivial.
10
that is, a non-chiral spectrum. The orbifold action reads
Θ : Ψ(z1 , z2 , z3 ) 7→ ±s2 Ψ(−z1 , −z2 , z3 )
Θ0 : Ψ(z1 , z2 , z3 ) 7→ s2 s3 Ψ(z1 , −z2 , −z3 )
(2.12)
and so we arrive at the following wavefunctions
- s2 > 0, s3 > 0
=⇒
Ψ = ψej2 ψej3 or ψoj2 ψoj3
=⇒
Iab = Ie2 Ie3 − Io2 Io3
- s2 > 0, s3 < 0
=⇒
Ψ = ψej2 ψoj3 or ψoj2 ψej3
=⇒
Iab = Ie2 Io3 − Io2 Ie3
- s2 < 0, s3 > 0
=⇒
Ψ = ψoj2 ψej3 or ψej2 ψoj3
=⇒
Iab = Io2 Ie3 − Ie2 Io3
- s2 < 0, s3 < 0
=⇒
Ψ = ψoj2 ψoj3 or ψej2 ψej3
=⇒
Iab = Io2 Io3 − Ie2 Ie3
where the relative minus sign in the expression for Iab comes from the fact that the two
fermions in (2.11) have opposite 4d chirality. Again, in each of the four cases above we
1
find that the index of net chirality Iab matches the expression (2.10) with Iab
= 0.
To summarise, by looking at the action of the orbifold on the open string degrees of
freedom we can recover the chiral index obtained in [16] via topological methods. While
we have focused on the fermionic modes, the same result is obtained by looking at the light
scalars at the D6-brane intersections. The method used here to compute the chiral index
Iab is perhaps more involved that the one in [16], but it also carries more information.
6
T
First, for the case where Iab
= 0 not only does it compute the net chiral index (2.10) but
also detects massless particles of opposite chirality that contribute with opposite signs
to the index.5 Second, this method not only gives the 4d massless spectrum, but also
the explicit expression for the open string wavefunctions in the internal dimensions of
the compactification. As we will now see, this will be crucial for studying in detail the
discrete flavor symmetries that appear in this class of models.
3
Discrete flavor symmetries for intersecting branes
Having reviewed D-brane models on T6 /Z2 × Z02 and how family replication arises for
them, we now turn to show the emergence of discrete flavor symmetries in such models.
5
This spectrum is only computed at tree-level in the string coupling gs , so one expects that vector-
like pairs of massless particles will gain a mass by means of quantum corrections, unless some discrete
symmetry forbids such mass term. See section 5 for an example.
11
More precisely, we will describe how the Dihedral group D4 and tensor products of it arise
in this context, and how the different families transform non-trivially under them.
As discussed in [8], D4 and other non-Abelian discrete flavor symmetries naturally
arise in the context of D-brane models, and in particular for models of magnetized Dbranes on T2n . There one can detect discrete gauge flavor symmetries in terms of a 4d
effective Lagrangian obtained via dimensional reduction. As shown in appendix B such
Lagrangian can also be obtained from models of intersecting D-branes on T2n , and so
in principle one can apply the 4d methods of [8] to detect discrete gauge symmetries.
However, it turns out that in models of intersecting D-branes the presence of discrete
flavor symmetries can be detected geometrically as well. This is particularly useful to
describe them in T6 /Z2 × Z02 , where applying dimensional reduction is not obvious for
certain sectors. In the following we will apply such geometric approach first for a toroidal
background and then for T6 /Z2 × Z02 . Finally, this geometric picture allows to quickly
detect when there is an exact discrete flavor symmetry and when such symmetry is just
approximate, as we briefly discuss.
3.1
Flavor symmetries on the torus
Let us consider type IIA string theory compactified on T6 = (T2 )1 × (T2 )2 × (T2 )3 . Such
manifold contains 6 continuous isometries (xi , yi ) → (xi + λxi , yi + λyi ), i = 1, 2, 3 which,
upon dimensional reduction of the metric, manifest as a U (1)6 gauge group in the 4d
effective theory. We will represent such 4d gauge bosons respectively as Vµxi and Vµyi .
Let us now introduce a D6-brane wrapping a factorizable three-cycle Πa of the form
(2.1). Geometrically, it is clear that the presence of such three-cycle breaks the invariance
under translations along the three directions of T6 transverse to the D6-brane worldvolume, while in the three directions parallel to Πa the translational isometries remain
unbroken. From the effective field theory viewpoint three generators of the initial U (1)6
gauge group become massive via a Stückelberg mechanism, in which the D6-brane scalars
φia that parametrize the transverse displacement of Πa in (T2 )i are eaten by the generators
of the corresponding isometry. Following appendix B, the Stückelberg Lagrangian reads
3
LSt = −
2
1X
∂µ φia − mia Vµxi + nia Vµyi
2 i=1
12
(3.1)
and so the bulk gauge symmetry U (1)6 is broken down to U (1)3 ×Zq1 ×Zq2 ×Zq3 . The U (1)
factors are generated by the massless combinations nia Vµxi + mia Vµyi , while the factors Zqi
are the discrete remnants of the broken U (1) symmetries generated by nia Vµyi − mia Vµxi .
This symmetry breaking pattern is similar to the one studied in [9], section 2.5, from
where one deduces that qi = (nia )2 + (mia )2 .
Needless to say, adding more D6-branes will further break the translational symmetry.
In particular, one would expect that by adding a D6-brane on a three-cycle Πb that
intersects Πa transversally all continuous symmetries are broken. Indeed, one then finds
that the Lagrangian reads
3
LSt
2
2
1X
= −
∂µ φia − mia Vµxi + nia Vµyi + ∂µ φib − mib Vµxi + nib Vµyi
2 i=1
(3.2)
i
and so if Iab
= nia mib − nib mia 6= 0 ∀i then all gauge bosons Vµxi , Vµyi , become massive. In
fact, as discussed in appendix B the remaining discrete gauge symmetry is given by
1 × ZI 2 × ZI 3
TTab6 = ZIab
ab
ab
(3.3)
Finally, additional D6-branes on three-cycles Πc , Πd , etc may further break this symmetry.
While understanding discrete symmetries from the viewpoint of the effective theory
is quite powerful, it is quite instructive to develop a more geometrical picture of their
meaning. For this let us focus on one of the T2 factors of T6 . As shown in figure 3a,
the absence of D-branes implies a U (1)2 gauge symmetry that corresponds to invariance
of the background upon translation in the x and y coordinates. Adding a D-brane a on
a 1-cycle n[a] + m[b] partially breaks this translational symmetry (fig. 3b): infinitesimal
−
translations in the direction →
v k = (n, m) leave the geometry invariant while those along
→
−
−
v = (−m, n) do not. Nevertheless, finite translations along →
v do leave the geometry
⊥
⊥
invariant and these, upon quotienting by the coordinate identifications of T2 , generate a
discrete group Zq with q = n2 + m2 . One then obtains a gauge group U (1) × Zq . Adding
a second D-brane b that intersects the first one (fig. 3c) will totally break the invariance
under infinitesimal translations. Still, a discrete translational symmetry remains, given by
the cyclic permutation of the intersection points of the two D-branes, and this generates
a ZIab gauge symmetry. Applying this result to each (T2 )i factor of T6 we obtain (3.3).6
6
In general, given a T2n geometry we have a U (1)2n translational symmetry. If we introduce two n-
13
y
y
y
a)
c)
x
y
x
b)
x
d)
x
Figure 3: T2 /Z2 with a) no branes b) one brane on the cycle (1,3) c) two branes on (1,3)
and (1,1) d) three branes on (1,3), (1,1) and (1,-1).
From this geometrical perspective one can also see that we are indeed dealing with a flavor
symmetry, that acts on the intersection points of each T2 as the shift generator


1






1




.
.
gT = 
.






1 


1
(3.4)
Finally, this symmetry is further broken if we include additional D-branes (fig. 3d). One
can check that if we add a D-brane c then the fundamental region of T2 will be divided
into d identical regions, with d = g.c.d.(Iab , Ibc , Ica ) [34]. The remaining discrete gauge
symmetry is then Zd , which corresponds to the common factor ZIab ∩ ZIbc ∩ ZIca of the
cycles Πna , Πnb that are each a Tn ⊂ T2n and that intersect transversally, then the group of translational
symmetry is broken to T = Γ/Γ̂, where Γ is the lattice generated by the intersection points and Γ̂ is the
lattice of coordinate identifications that defines T2n . When T2n is factorizable T is a direct product of
discrete subgroups, as in (3.3).
14
symmetries for each pair of D-branes. Going back to the case of T6 = (T2 )1 ×(T2 )2 ×(T2 )3 ,
we conclude that for a system of three D6-branes the translational symmetry is given by
TTabc
= Zd1 × Zd2 × Zd3
6
(3.5)
i
i
i
). This kind of symmetries will constrain the values of the
, Ica
, Ibc
with d i = g.c.d.(Iab
Yukawa couplings of this sector, as pointed out in [8, 21, 34].
In fact, the above is not the complete flavor symmetry of the model, as there are
further bulk symmetries that are broken by the presence of the D6-branes. Besides the 4d
U (1)6 gauge symmetry arising from the metric there will be a 4d U (1)6 gauge symmetry
that comes from the B-field, and is generated by the 4d gauge bosons Bµxi , Bµyi that arise
upon dimensional reduction. From appendix B, the Stückelberg Lagrangian for a single
D6-brane reads
3
LSt
2
1X
= −
∂µ ξai − nia Bµxi − mia Bµyi
2 i=1
(3.6)
with ξai the Wilson line modulus of the D6-brane on (T2 )i . This action also has a simple
geometrical interpretation, namely that acting with a B-field gauge generators induces a
Wilson line on the D6-brane via pull-back on its worldvolume Πa . A gauge transformation
along −mia Bµxi + nia Bµyi will have vanishing pull-back and will remain a symmetry of the
background, while one along nia Bµxi + mia Bµyi will be detected by the D6-brane and the
corresponding U (1) symmetry will be broken to a discrete subgroup. One can again see
that the remaining symmetry is given by U (1)3 × Zq1 × Zq2 × Zq3 .
Adding further D6-branes will generalize this Lagrangian to
3
LSt
2
1 XX
∂µ ξαi − niα Bµxi − miα Bµyi
= −
2 α i=1
(3.7)
with α = a, b, c, . . . . For a system of two D6-branes a and b the symmetry is broken to
1 × ZI 2 × ZI 3
WTab6 = ZIab
ab
ab
(3.8)
which in principle looks similar to (3.3) but the action of the generators on the flavor
degrees of freedom is quite different. In this case the generator of the flavor symmetry
15
acts on the intersection points of each T2 as the clock generator7


1




1
2πi


e N




2
2πi N
gW = 

e




...




N −1
e2πi N
(3.9)
i
. As it is easy to check the generators (3.4) and (3.9) do not commute, and
with N = Iab
so with their combined action they end up generating the discrete non-Abelian group of
the form HN ' (ZN × ZN ) o ZN for each T2 , or more precisely
1 × HI 2 × HI 3
Pab
T6 = HIab
ab
ab
(3.10)
which is the result obtained in the T-dual picture of magnetized D9-branes [8,21]. Finally,
for a triplet of D6-branes this symmetry is reduced to
Pabc
T6 = Hd1 × Hd2 × Hd3
(3.11)
i
i
i
with again d i = g.c.d.(Iab
, Ibc
, Ica
)
3.2
Flavor symmetries on Z2 × Z02
Let us now consider the case of type IIA string theory compactified on T6 /Z2 ×Z02 . Unlike
the case of T6 the orbifold background does not have any continuous isometry even in the
absence of D-branes. Hence the dimensional reduction that led to effective actions of the
form (3.2) or (3.7) does not apply, and we need to use a different method to determine
which are the discrete flavor symmetries that can arise in this case. Notice that the
same will be true in Calabi-Yau compactifications, as these manifolds do not contain any
continuous isometry either.
Fortunately in our discussion of T6 we have developed an alternative method for
detecting discrete flavor symmetries. For instance, in the case of translational isometries
7
The action of Bµ is equivalent to switching on a Wilson line, Aα = dχα = 2πdζ α , where ζ α ∼ ζ α + 1
is the coordinate of the D6-brane α along the corresponding T2 . An open string located at ζ α = j/N
α
and with charge qα will have its phase shifted as eiqα χ = e2πiqα j/N , from where the action (3.9) follows.
16
the flavor symmetry was understood as the group of isometries of the manifold that is
also preserved by the D-brane configuration. This observation can also be applied to the
T6 /Z2 × Z02 orbifold, whose group of translational isometries is discrete and given by Z26 .
Upon dimensional reduction this will give rise to a 4d discrete Z26 gauge group that will
be broken to a subgroup by the inclusion of D6-branes, and this subgroup will be part of
the discrete flavor symmetry of the model.
To get an idea of this symmetry breaking let us again consider the toy example T2 /Z2 .
The Z2 quotient is generated by z 7→ −z and so there are four fixed points that break the
continuous isometry group U (1)2 of T2 down to Z2 × Z2 . The generators of this discrete
group are the actions z 7→ z + 1/2 and z 7→ z + τ /2, with τ the complex structure of the
torus, that interchange the fixed points at {0, 1/2, τ /2, (1 + τ )/2} among them.
y
y
3
7
6
z→z+
5
4
2
1+τ
2
2
1
1
3
4
0
3
7
2
1
0
6
13
4
4
x
Figure 4: Translation z → z +
1+τ
2
52
x
in a square torus. This is the generator of the shift
symmetry in the intersecting brane picture.
Let us now introduce D-branes in this background. In our toy example fractional Dbranes are represented by 1-cycles that pass through two of the four fixed points of T2 /Z2 .
It is then clear that the presence of a single D-brane breaks the group of translations
Z2 × Z2 down to Z2 , where this latter Z2 interchanges the two fixed points that the
D-brane goes through. For instance, as shown in figure 4 a D-brane whose wrapping
numbers (n, m) are both odd will go through the fixed points {1,4} or {2,3} (see table 1).
The symmetry of this system is then the Z2 generated by z → z +
1+τ
2
that interchanges
the fixed points as 1 ↔ 4 and 2 ↔ 3. As figure 4 also shows this Z2 symmetry will still
17
be preserved after we introduce a second D-brane, provided that it also goes through the
fixed points {1,4} or {2,3} or, in other words, if its wrapping numbers (n, m) are both odd
as well. In general, a pair of fractional 1-cycles on T2 /Z2 will preserve a Z2 translational
symmetry if they belong to the same row of table 1, which is equivalent to asking that
the intersection number Iab = na mb − nb ma is even. Finally, three or more 1-cycles will
preserve the same Z2 symmetry if they all belong to the same row of table 1, or in other
words if all the pairwise intersection numbers Iab , Ibc , Ica , . . . are even.
One may now generalize these observations to the case of (T2 )1 ×(T2 )2 ×(T2 )3 /Z2 ×Z02 ,
as each (T2 )i factor will behave like our toy example. Instead of our previous result (3.3)
for a pair of D6-brane on T6 we now have that each (T2 )i factor contributes at most with
i
a Z2 symmetry, and only if the intersection number Iab
is even. Hence
TTab6 /Z2 ×Z02 = Zρ1 × Zρ2 × Zρ3
(3.12)
with ρi defined as in (2.8). Similarly, for a system of three D6-branes we have
TTabc
6 /Z ×Z0 = Zd1 × Zd2 × Zd3
2
2
(3.13)
i
i
i
where now d i = g.c.d.(2, Iab
, Ibc
, Ica
).
Just like in the case of T6 , this will not be the whole flavor symmetry group. There
will also be a symmetry group generated by discrete gauge transformations of the B-field,
whose gauge bosons Bµxi ,yi are projected out infinitesimally by the orbifold. Just like for
finite translations, these finite B-field transformations generate a Z26 group on T6 /Z2 × Z02
that is broken to a subgroup when D6-branes are introduced. One can check that this
subgroup WTab6 /Z2 ×Z0 is isomorphic to (3.12) when two D6-branes are introduced, and
2
similarly for
WTabc
6 /Z ×Z0
2
2
and (3.13) for a triplet of D6-branes. As in the T6 case the two
Z2 subgroups that arise from (T2 )i do not commute, but rather generate the non-Abelian
group (Z2 × Z2 ) o Z2 ' H2 , which is nothing but the Dihedral group D4 . The final
symmetry group for a pair of D6-branes would then be given by
[ρ −1]
1
Pab
T6 /Z2 ×Z02 = Hρ1 × Hρ2 × Hρ3 = D4
[0]
[ρ −1]
× D4 2
[ρ −1]
× D4 3
(3.14)
[1]
where D4 is the trivial group and D4 = D4 , while for a D6-brane triplet we should have
[d −1]
1
Pabc
T6 /Z2 ×Z02 = D4
[d −1]
× D4 2
18
[d −1]
× D4 3
(3.15)
i
i
i
with d i = g.c.d.(2, Iab
, Ibc
, Ica
).
We will rederive this result in the next section, where we will use the T-dual framework
of magnetized D-branes to obtain (3.14) and (3.15), as well as to make contact with the
results of [19–22]. In addition to deriving the symmetry group we will use the magnetized
picture to classify under which representations do the chiral families transform on each
model. The reader not interested in such details may find the results summarized below,
and may safely skip to section 5 where they are applied to specific examples.
Summary
In general we find that the wavefunctions that correspond to bifundamental fields (Na , N̄b )
are of the form
j1 ,j2 ,j3
j1
j2
j3
ψab
= ψab
· ψab
· ψab
(3.16)
ji
i
where ψab
lives in (T2 )i . Depending on the signs of the intersection numbers Iab
these
wavefunctions will be even or odd under the action zi 7→ −zi , as discussed below (2.9).
ji
is even the index ji will run over Iei values and if it is odd over Ioi values, with Iei
If ψab
i
is even, this index will transform under a specific
and Ioi defined in (2.7). Moreover, if Iab
representation of the flavor symmetry group D4 of (T2 )i . This representation will depend
i
on the value of Iab
and the wavefunction parity, as shown in table 2.
i
|
|Iab
ji
ψeven
ji
ψodd
i
dim = |Iab
|/2 + 1
s+1
s+1
8s + 4
s+1
s+1
⊕ R2
s
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
s+2
8s + 8
s
⊕ R2
4s + 2
s+1
s+1
i
dim = |Iab
|/2 − 1
s+1
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
s
s
s
s+1
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
s
s+1
s+1
s+1
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
Table 2: Different family representations under the flavor symmetry group D4 on each
(T2 )i . Here R2 stands for the 2-dimensional irreducible representation of the dihedral
group D4 , see (4.20), while (±, ±0 ) stands for the one-dimensional representation in which
the two generators of D4 act respectively as ±I and ±0 I.
19
3.3
Exact versus approximate symmetries
The previous discussion is quite useful in order to draw a notion of exact and approximate
discrete symmetry for this class of models. By exact symmetry it is meant a discrete gauge
symmetry of the 4d effective field theory, in the sense of [35–42]. The non-Abelian discrete
symmetries discussed in [8] are of this sort, the procedure to detect the gauge nature of a
discrete symmetry being the construction of the effective 4d Lagrangian. For the case of
the Z2 × Z02 orbifold, the construction of such 4d Lagrangian is beyond the scope of this
paper, and so we will instead adopt a different approach and discuss the exactness of a
discrete flavor symmetry by means of the geometric intuition developed above.
As pointed out in the last subsection the Z2 × Z02 orbifold background has a group
of translational isometries given by TZ2 ×Z02 = Z26 . Coming from isometries of the internal
manifold, this group naturally translates as a Z26 gauge group in the 4d effective theory.
Adding a fractional D6-brane will break this group down to Z23 , where each Z2 factor
comes from a different (T2 )i . This Z23 symmetry group can be understood as the group
of translations that leaves both the orbifold background and the D-brane invariant, and
so it is a natural candidate for a 4d discrete gauge symmetry of the orbifold plus D-brane
background. The question is now if the whole set of D-branes in a given model will respect
such symmetry as well, or in other words if the whole orbifold plus D-brane backgrounds
will be invariant under this Z23 translational symmetry or a subgroup thereof.8
By looking at figure 4 it is clear that a group of two or more D-branes will be invariant
under a Z2 shift symmetry of (T2 )i if all of them go through the same pair of fixed points.
In fact, the condition for the symmetry to be exact is weaker, and we only need to require
i
that all the intersection numbers Iαβ
between D-branes in this two-torus are even. The
same is true for the Z2 discrete symmetry that arises from the B-field, and so for the whole
(i)
D4 that both Z2 actions generate. Let us denote as D4 the dihedral flavor symmetry
that may arise from (T2 )i , we then have that
(i)
(i)
D4 is exact ⇐⇒ Iαβ is even ∀α, β
8
(3.17)
Another important element of a D-brane model is the orientifold planes or O-planes, which we have
so far ignored. One can check that their presence does not further break these discrete symmetries, at
least for the class of models with rectangular (T2 )i that we will consider in section 5.
20
and so the group (3.15) may become a gauge or exact discrete gauge symmetry of the 4d
effective theory depending on the whole set of D-brane intersection numbers.9
It may seem that the exactness condition (3.17) is kind of restrictive when constructing
explicit D-brane models. However, as follows from the results of [16], one typically needs
that D6-branes go through the same fixed points in order to satisfy the RR twisted tadpole
conditions necessary to construct an anomaly-free consistent model. From this viewpoint,
the stringy consistency conditions of the model render natural the appearance of exact
discrete flavor symmetries in the low energy theory, as the examples of section 5 illustrate.
To be more precise let us consider a Z2 × Z02 model with K stacks of fractional
D6-branes, and let us separate them in two subgroups A = {a1 , a2 , a3 , . . . } and B =
{b1 , b2 , . . . }, wrapping three-cycles Πai , Πbj of the orbifold. The group A will yield a 4d
Q
gauge group i U (Nai ), whose chiral spectrum will be specified by the intersection numbers Iak al . Typically, demanding that this sector of the theory is free of chiral anomalies
by itself will impose cancellation of RR twisted tadpoles within the group A of D6-branes,
(i)
and this will most likely happen when all the D6-branes in A respect the same D4 symmetry in (T2 )i . There will be then a discrete symmetry group of the form (3.15) acting
on this sector and constraining its couplings in the 4d effective field theory.
We may in particular consider the case where the group A of D6-branes contains the
spectrum of the Standard Model or an extension thereof, while the group B of additional
D6-branes contains an extra (hopefully hidden) sector of the theory. If all the (T2 )i inter(i)
section numbers Iai k al are even there will be a flavor symmetry group D4 acting on the
visible sector of the theory. However, the extra sector B may not respect such symmetry,
(i)
and if there is a single intersection number Iai k bl which is odd then D4 will not be an
exact symmetry of the model. Nevertheless, it can still be considered an approximate
symmetry of the sector A, because all the couplings within this sector must still respect
this symmetry at least at tree-level. In particular, in a supersymmetric model the holomorphic Yukawa couplings of this sector will be constrained by the flavor symmetry at all
9
(i)
Strictly speaking if (3.17) is true then D4 is a global symmetry of the 2d action of the BCFT theory,
which becomes a local symmetry in target space to all orders in string perturbation theory. One should
still check that this symmetry is preserved at the non-perturbative level in the string coupling, something
that here is assumed. We leave a more detailed analysis of this subject for future work.
21
orders in perturbation theory. The Kähler potential, on the other hand, may already get
symmetry-breaking corrections at the perturbative level by effects involving the D-branes
bl that do not respect the symmetry (e.g., massive open string attached to bl running in
loops). It would be interesting to see if the scenarios and techniques that apply to approximate continuous symmetries, see e.g. [43,44], could also be at work for this case. We
leave for the future to explore the phenomenological consequences of these approximate
symmetries in realistic and semi-realistic D-brane models.
4
Discrete flavor symmetries for magnetized branes
An interesting feature of the D-brane models analyzed in the previous sections is that they
have a well-known T-dual description in terms of magnetized D-branes, which is a fruitful
arena for understanding flavor symmetries. Indeed, as emphasized in [24], the framework
of magnetized D-branes allows to compute 4d effective couplings by first solving for the
chiral modes internal wavefunction profile, and then calculating their overlap over the
extra dimensions of the compactification. As pointed out in [21], by inspection of these
zero mode wavefunctions one can understand the flavor symmetries present in the model.
Finally, it was shown in [8] how to obtain from this framework a 4d effective Lagrangian
describing such discrete gauge flavor symmetries.
In the following we will rederive our previous results in the dual context of magnetized
D-brane models, both in T6 and in the Z2 × Z2 orbifold. This will allow to perform a
more systematic analysis of the discrete flavor symmetries, and in particular to see under
which representation transform the different families of chiral multiplets.10
10
Although more systematic, the field theory framework of magnetized D9-branes is less general than
the framework of intersecting D6-branes, because it fails to capture the actual 4d effective theory when
the magnetic fluxes are not diluted and/or when anti-D9-branes or D-branes of lower dimension are
present. In this sense, the results of this section can be seen as complementary to the ones obtained
previously in the context of intersecting D6-branes.
22
4.1
Non-Abelian flavor symmetries from magnetization
Let us first consider type IIB string theory compactified on the factorized six-torus T6 =
(T2 )1 × (T2 )2 × (T2 )3 and N D9-branes filling the whole of 10d space-time. We may
add an non-trivial magnetization hF2 i along the coordinates of T6 without breaking 4d
Poincaré invariance. In particular, we may choose a U (N ) Yang-Mills field strength of
the form

mia
I
 nia Na

3
X
πi 

F2 =
i 
Im
τ

i=1


mib
I
nib Nb
mic
I
nic Nc
...


 i
 dz ∧ dz̄ i



where z i = dxi +τ i dy i is the complexified coordinate of (T2 )i , Nα = n1α n2α n3α , N =
(4.1)
P
α
Nα .
Each block within (4.1) can be seen as a different D9-brane with ‘magnetic numbers’
niα , miα ∈ Z and with gauge group U (d1α d2α d3α ), dαi = g.c.d.(niα , miα ) [45]. One can then
describe a pair of D9-branes in terms of these magnetic numbers
D9a : (n1a , m1a ) (n2a , m2a ) (n3a , m3a )
D9b : (n1b , m1b ) (n2b , m2b ) (n3b , m3b )
(4.2)
in a rather analogous fashion to (2.3). In fact, both configurations are mapped to each
other by performing three T-dualities, as have been used extensively in the literature. In
this correspondence, the matter localized at the D6-brane intersections Πa ∩ Πb is mapped
to the set of zero modes that arise from a Na × Nb submatrix of the 10d U(N) adjoint
fields (Ψ, AM ) [24]. In the following we will assume that dαi = g.c.d.(niα , miα ) = 1 and, in
particular, that niα = 1 ∀α, i.11 This greatly simplifies the analysis, since then Nα = 1
and the internal profile of the 4d chiral zero modes is an scalar wavefunctions ψ j instead
of a matrix of wavefunctions.
As pointed out in [21], by inspection of such zero mode wavefunctions one can guess
the flavor symmetry of a model of magnetized D-branes. However, as we will now show,
one can directly characterize this flavor symmetry group by looking at the symmetries of
11
While more involved, one can generalize the analysis for the case niα > 1, along the lines of [23, 24].
The case where some of the niα = 0 also makes sense, and describes a model with D7, D5 or D3-branes.
This case, however, it is difficult to analyze from the field theory viewpoint and it is then more convenient
to analyse the discrete flavor symmetries from the T-dual framework of intersecting D6-branes.
23
the D-brane configuration, without solving for any zero mode. For simplicity, let us first
consider a T2 and a U (2) gauge sector with a magnetization


m
πi  a
 dz ∧ dz̄
F2 =
Im τ
mb
(4.3)
that breaks the gauge symmetry down to U (1)a × U (1)b . In general this system is interpreted as two magnetized D-branes a and b, whose zero modes in the ab sector feel the
2
T
relative flux M = ma − mb = −Iab
. Even if F2 is constant we have a vector potential of
the form

A(x, y) = π 

ma
mb
 (xdy − ydx)
(4.4)
which breaks the invariance under translations. More precisely we have that
A(x + λx , y) = A(x, y) + πλx (ma Xa + mb Xb )dy,
where

Xa = 
(4.5)
A(x, y) − πλy (ma Xa + mb Xb )dx
A(x, y + λy ) =
1
0



and Xb = 

0
1

(4.6)
Eq.(4.5) can be interpreted as the fact that, in the presence of the background flux (4.3),
an arbitrary translation is no longer a symmetry of the theory because hAi is not invariant
under it. Nevertheless, from (4.5) we see that this variation is equivalent to a linear gauge
transformation, which can in turn be interpreted as a Wilson line. For certain discrete
choices of λx , λy such Wilson line will be trivial, and this will correspond to a discrete
symmetry of the configuration.
To properly see this point let us replace the gauge potential A by a gauge covariant
object such as the covariant derivative iD. In addition, we must take into account that
in a gauge theory translations of the form x → x + λx are generated by exp(λx Dx ). The
gauge covariant version of (4.5) is then
eλj Dj iDk e−λj Dj = iDk + λj Fjk
(4.7)
where j, k = x, y, and we have used that [Dj , Dk ] = −iFjk . In fact, translations are not
the only possible gauge transformations that we can perform but, just like in the case of
24
D-branes at angles, there are also the ones generated by the 4d gauge bosons Bµx,y that
arise from the B-field. These act on the covariant derivative as a diagonal linear gauge
transformation, namely
eµj Bj iDk e−µj Bj = iDk + µj δjk I2 ,
Bx = 2πi x I2 , By = 2πi y I2
(4.8)
Finally, we can write both (4.7) and (4.8) in the form
eλx Dx +µy By iD e−λx Dx −µy By = Ξy iD Ξ−1
y ,
Ξy = e2πi(ξy,a Xa +ξy,b Xb )y
eλy Dy +µx Bx iD e−λy Dy −µx Bx = Ξx iD Ξ−1
x ,
Ξx = e2πi(ξx,a Xa +ξx,b Xb )x
(4.9)
where
ξy,a = λx ma + µy
ξx,a = −λy ma + µx
ξy,b = λx mb + µy
(4.10)
ξx,b = −λy mb + µx
(4.11)
with ξx,α representing a Wilson line for the gauge group U (1)α along the coordinate x,
and similarly for ξy,α . Notice that no Wilson lines are induced for U (1)a if µy = −λx ma
and µx = λy ma , and so this gauge sector remains invariant under this particular combined
action of the bulk gauge transformations. In other words, the magnetized D-brane α = a
breaks the original U (1)4 symmetry of the bulk down to (U (1) × Zq )2 , similarly to the
previous case of a D-brane wrapping a 1-cycle. A similar statement can be made for the
D-brane b, and it can all be encoded in the 4d effective field theory via the following
Stückelberg Lagrangian
LSt = −
2
2 o
1 Xn
∂µ ξx,α + mα Vµy − Bµx + ∂µ ξy,α − mα Vµx − Bµy
2 α=a,b
(4.12)
which is a particular case of (B.16), derived in appendix B from dimensional reduction.
Here ξx,α , ξy,α represent the 4d scalar fluctuations corresponding to the Wilson lines of
U (1)α , Vµx,y are the gauge bosons that arise from the metric and Bµx,y from the B-field.
One can now interpret (4.9) as advanced before: whenever (ξy,a , ξy,b ) ∈ Z2 we have a
trivial Wilson line shift in the rhs of (4.9), and so the corresponding gauge transformation
generated by a Vµx and Bµy is a symmetry of the system. One can check that there
are M = ma − mb inequivalent values of (λx , µy ) that correspond to (ξy,a , ξy,b ) ∈ Z2 ,
and that such values generate a residual ZM symmetry. Similarly, there are M values
25
of (λy , µx ) such that (ξx,a , ξx,b ) ∈ Z2 , and these generate an additional ZM symmetry.
Finally, because of (4.7) these two ZM symmetries do not commute, and we end up with
a non-Abelian symmetry group given by HM ' (ZM × ZM ) o ZM .
It is instructive to apply this discrete symmetry to the chiral zero mode wavefunctions
of this magnetized system, and in particular to those in the bifundamental representation
(+1, −1) of U (1)a × U (1)b , which is where families of chiral matter arise from. On these
modes Bx,y act trivially, so the above discrete symmetry is implemented by12
nx
e M Dx
ny
and e M Dy
nx , ny = 0, . . . , M − 1
(4.13)
with these operators acting on the zero modes obtained by solving the internal Dirac or
Laplace equations on T2 [24]



j





eiπM zImz/Imτ ϑ  M  (M z, M τ )



0
ψ j,M (z, z̄) =



j




eiπM z̄Imz̄/Imτ ϑ  M  (M z̄, M τ̄ )



0
if M > 0
(4.14)
if M < 0
j = 0, 1, . . . , |M | − 1 running over independent zero mode solutions. One can check that
nx
nx
gW
= e M Dx ψ j,M = e2πi
nx j
M
ψ j,M
ny
n
gT y = e M Dy ψ j,M = ψ j+ny ,M
so that if we consider the vector of wavefunctions

ψ 0,M

..

Ψ = 
.

ψ M −1,M
(4.15)





(4.16)
we have that the group elements gT and gW act as (3.4) and (3.9) respectively, generating
the discrete Heisenberg group HM ' (ZM × ZM ) o ZM as mentioned above.
12
In [8] the alternative set of operators was considered
nx
e M Xx
e
ny
M
Xy
Xx = ∂x − πi (ma Xa + mb Xb ) y
Xy = ∂y + πi (ma Xa + mb Xb ) x
in order to implement the action of the flavor symmetry group on wavefunctions. Both choices are in
fact equivalent as they differ by a trivial Wilson line shift.
26
If we now consider the full T6 = (T2 )1 × (T2 )2 × (T2 )3 magnetized D9-brane system,
we obtain that the zero mode wavefunctions for the D9a D9b sector are [24]
1
j ,−Iab
j1 ,j2 ,j3
= ψab1
ψab
2
j ,−Iab
(z1 , z̄1 ) · ψab2
3
j ,−Iab
(z2 , z̄2 ) · ψab3
(z3 , z̄3 )
(4.17)
i
= mib − mia and ψ j,M as in (4.14). The number of zero modes in the ab sector is
with Iab
6
3
2
1
T
| , as expected from the T-dual intersecting D6-brane system,
||Iab
||Iab
| = |Iab
given by |Iab
6
T
and their 4d chirality is again given by sign(Iab
). From our discussion on T2 it follows
i . We then obtain
that each index ji transforms in the fundamental representation of HIab
1 × HI 2 × HI 3 ,
again that the flavor symmetry group of this sector is given by Pab
T6 = HIab
ab
ab
as in (3.10).
Let us now consider magnetized D9-branes in a T6 /Z2 × Z2 orbifold background,13
again with the Z2 generators acting as (2.2). Because of the presence of the orbifold fixed
loci, the U (1)6 translational symmetry of T6 is broken down to the discrete subgroup
Z62 , and this reduces the set of operators of the form (4.13) that are compatible with the
symmetries of the background.
For our purposes it is instructive to again consider the toy example T2 /Z2 with
Z2 action generated by z 7→ −z. As before, this background has four fixed points at
{0, 1/2, τ /2, (1 + τ )/2} that are interchanged by the Z2 × Z2 symmetry group generated
by the discrete translations z 7→ z + 1/2 and z 7→ z + τ /2. Let us now consider the magnetized U (2) sector (4.3) in this background. The T2 /Z2 discrete isometry z 7→ z + 1/2 is
implemented by exp( 21 Dx ), while z 7→ z + τ /2 is implemented by exp( 12 Dy ). From the discussion above, we know that these operators correspond to symmetries of the magnetized
system only if they belong to (4.13), or in other words if M is even. We then find that a
pair of magnetized D-branes respects the orbifold translational Z2 × Z2 symmetry if and
only if Iab = even, exactly as we found in the T-dual picture of intersecting D-branes.
M/2
As it is clear from (4.15), for M even the group elements gT
M/2
and gW
will generate
a discrete flavor symmetry group acting on the zero mode wavefunctions. Because the
13
More precisely, we consider the type IIB orbifold background mirror symmetric to our previous type
IIA T6 /Z2 × Z02 background. These two backgrounds are quite similar but not exactly the same, because
upon three T-dualities the choice of discrete torsion of a Z2 ×Z2 orbifold is reversed. As a result, the fixed
points of the type IIB Z2 × Z2 orbifold considered in this section contain collapsed two and four-cycles
instead of collapse three-cycles.
27
group action is non-Abelian and in general it describes a discrete Heisenberg group, we
can identify the flavor group with H2 ' (Z2 × Z2 ) o Z2 ' D4 . As we will see, the families
of wavefunctions indeed arrange themselves in representations of the dihedral group D4 .
4.2
Wavefunction representations in T6 /Z2 × Z2
Let us consider in detail the wavefunctions for bifundamental fields in a magnetized D9brane model in T6 /Z2 × Z2 . In general the spectrum of zero modes will be similar to the
case of T6 , except that we need to project out those modes that are not invariant under
the orbifold action. The procedure for finding the surviving chiral families works pretty
much like in section (2.2) (see [19] for a previous discussion). The chiral matter will have a
specific SO(8) weight representation inherited from the 10d fields (Ψ, AM ). In particular,
i
for 4d massless fermions we also find the representation (2.5) with si = sign(Iab
). The
orbifold generators will act on the internal Lorentz indices of the chiral fermions as in
(2.6) and select either even or odd linear combinations of wavefunctions for each (T2 )i .
We will again have wavefunctions of the form (4.17) but with (4.14) replaced by
j
ψeven
∝ ψ j,M + ψ M −j,M
j
ψodd
∝ ψ j,M − ψ M −j,M
or
(4.18)
depending on each case and (T2 )i . The family indices ji will run over Iei or Ioi different
values, cf.(2.7), and so the total number of chiral families will again be given by (2.10).
In the following we will analyze the different representations of the flavor group under
j1 j2 j3
j1 j2 j3
which the chiral families transform. Since each index in ψab
= ψab
ψab ψab transforms
ji
independently we can treat each wavefunction factor ψab
separately, which is equivalent
to consider the representations of even and odd wavefunctions in our toy example T2 /Z2 .
As we now show, the decomposition into irreducible representations of D4 depends on the
value of the T2 magnetic flux M , which we assume an even number
M=2
In this case we have that

=2

ΨM
even =
28
ψ
0,2
ψ
1,2


(4.19)
while all odd wavefunctions are projected out. The group elements gW and gT generate
the 2 × 2 irreducible representation of D4 , given by







-1 0
0
0 -1
1 0
 , M3 = 
 , M4 = 
 , M2 = 
M1 = 
0 -1
-1
1 0
0 1







0 1
1 0
-1 0
0
 , M8 = 
 , M6 = 
 , M7 = 
M5 = 
1 0
0 -1
0 1
-1
1
0
-1
0

 (4.20)


More precisely, we have that gW = M5 , gT = M6 and that all other elements are generated
by multiplication of these two. In the following we will refer to this 2-dimensional, faithful
irrep of D4 as R2 .
M=4
For M = 4 we have that

=4
ΨM
even


= 

ψ

0,4
√1 (ψ 1,4
2


+ψ ) 

3,4
1
=4
1,4
ΨM
− ψ 3,4 )
odd = √ (ψ
2
and
(4.21)
ψ 2,4
2
and TS = gT2 . On the three
and the generators of the group action are given by TC = gW
dimensional vector of even wavefunctions these elements read




1 0 0
0 0 1








TC =  0 −1 0 
and
TS =  0 1 0 




0 0 1
1 0 0
(4.22)
which is not an irreducible representation: we can consider a new basis of even wavefunctions

=4
Ψ̃M
even
0,4
2,4
ψ +ψ
1 
 1,4
= √  ψ + ψ 3,4
2
ψ 0,4 − ψ 2,4





(4.23)
in which TC = diag (1, −1, 1) and TS = diag (1, 1, −1). This representation of D4 is thus
equivalent to
(+, +) ⊕ (−, +) ⊕ (+, −)
29
(4.24)
where (−, +) is the one-dimensional representation of D4 such that TC = 1 and TS = −1.
One can also see that D4 acts on Ψodd as a one-dimensional representation such that
TC = TS = −1, or using the above notation as
(−, −) ' det R2
(4.25)
M=2k
For general M = 2k, k ∈ Z it is convenient to define the even wavefunctions as
ξe0,k = ψ 0,2k
1
ξej,k = √ (ψ j,2k + ψ 2k−j,2k )
2
k,k
k,2k
ξe
= ψ
for j = 1, 2, . . . , k − 1
(4.26)
One can check that the flavor group generators act on them as follows
TC ξej,k = (−1)j ξej,k
TS ξej,k = ξek−j,k
k
and TS = gTk . In matrix terms one obtains the following
where we have defined TC = gW
(k + 1)-dimensional representation.



1 0 ...

ξe0,k



 0 −1 . . .
 .. 
=2k
ΨM
≡
 .  , TC = 
even
 .. ..


 . .

ξek,k
0 0 ...


0 ...




 0 ...
0 
 , TS = 

 ..
..

 .
.


k
1 ...
(−1)
0
0 1
1
..
.
0
..
.
0 0




 (4.27)



One can see that for k = M/2 even the two generators TC and TS commute, so they
can be simultaneously diagonalized. This diagonalization corresponds to decompose the
original (k + 1)-dimensional representation into a sum of one-dimensional ones. In this
case one gets the following decomposition:
k = 4s
s
s
s
−→ k + 1 = ⊕s+1
i=1 (+, +)i ⊕j=1 (+, −)j ⊕k=1 (−, +)k ⊕l=1 (−, −)l
s+1
s+1
s
k = 4s + 2 −→ k + 1 = ⊕s+1
i=1 (+, +)i ⊕j=1 (+, −)j ⊕k=1 (−, +)k ⊕l=1 (−, −)l
(4.28)
where (1 , 2 ) stands for the one-dimensional representation in which TC = 1 and TS = 2 .
30
Let us now examine the case in which k = 2s + 1 is odd. The matrices TC and TS are
(2s+2)-dimensional and they do not commute in this case, but they are still diagonalizable
by blocks. Every block is the same and identical to the 2 × 2 matrices in (4.20), which
k
means that we can write TC = gW
and TS = gTk as ⊕s+1 M5 and ⊕s+1 M6 , respectively. In
other words the flavor group action can be expressed as
⊕s+1
i=1 (R2 )i
(4.29)
For the case of odd wavefunctions we can take the following definitions
1
ξoj,k ≡ √ (ψ j,2k − ψ 2k−j,2k )
2
for j = 1, 2, . . . , k − 1.
and check that in this case the group generators act as



−1 0 . . .
0

ξo1,k



 0 +1 . . .
0
 .. 
M =2k
Ψodd ≡  .  , TC = 
 ..
..
..


 .
.
.
k−1,k

ξo
0
0 . . . (−1)k−1
(4.30)


0 ...




 0 ...

 , TS = 
 ..

 .



−1 . . .
0
−1
−1
..
.
0
..
.
0








0
(4.31)
k
and TS = gTk . For k even these two matrices commute and one
Where again TC = gW
has an Abelian (k − 1)-dimensional representation decomposable as
s
s
s
−→ k − 1 = ⊕s−1
i=1 (+, +)i ⊕j=1 (+, −)j ⊕k=1 (−, +)k ⊕l=1 (−, −)l
k = 4s
k = 4s + 2 −→ k − 1 = ⊕si=1 (+, +)i ⊕sj=1 (+, −)j ⊕sk=1 (−, +)k ⊕s+1
l=1 (−, −)l
M
Ψeven
Ψodd
dim = M/2 + 1
s+1
s+1
8s + 4
s+1
⊕ R2
s
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
s+2
8s + 8
s+1
s+1
s+1
dim = M/2 − 1
s
⊕ R2
4s + 2
(4.32)
s+1
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
s
s
s
s+1
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
s
s+1
s+1
s+1
⊕ (+, +) ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
Table 3: Different family representations depending on the value of the magnetization
M ∈ 2Z for even and odd wavefunctions. Here R2 stands for the 2-dimensional irreducible
representation of the dihedral group D4 , as in (4.20).
31
Finally, for k = 2s + 1 odd the group generators do not commute, but just like in the
case of even wavefunctions the 2s-dimensional representation is reducible into s copies of
the two-dimensional representation R2 . We then have that the flavor group acts as
⊕si=1 (R2 )i
(4.33)
All these results have been summarized in table 3. In the next section we will apply them
to see how families of quarks and leptons transform in specific semi-realistic models.
5
Examples
In this section we illustrate our general analysis via a couple of semi-realistic examples.
More precisely, we will consider two intersecting D6-brane models on the Z2 × Z02 orbifold
with a Pati-Salam gauge group. The first example is a four generation model already constructed in [16], with a D4 × D4 × D4 symmetry group constraining its Yukawa couplings.
The second example is a new, three generation model with a D4 symmetry group.
One important ingredient of these models is the presence of orientifold planes, that
allow to construct consistent and stable D-brane configurations. While the presence of Oplanes does not change the discrete symmetries of a Z2 × Z02 orbifold background, it does
affect the D-brane content of a model and the associated 4d chiral spectrum. Hence, before
presenting our examples we briefly review the effect of adding an orientifold projection to
the Z2 × Z02 orbifold.
5.1
Orientifolding
In general, in order to build consistent, stable and 4d Poincaré invariant models based on
intersecting or magnetized D-branes in Calabi-Yau compactifications we need to include
the presence of negative tension objects that cancel the positive tension of the D-branes.
The simplest way to do so is to include the presence of the non-dynamical, negative tension
objects known as orientifold planes. In the case of type IIA string theory compactified on
(T2 )1 × (T2 )2 × (T2 )3 this is achieved by modding out the theory by ΩR, where Ω stands
for the worldsheet parity operator and R for the anti-holomorphic involution R : zi 7→ z̄i .
For this the D-brane configuration has to be invariant under the action of ΩR, and so for
32
each D6-brane wrapping the three-cycle (2.1) there must be another D6-brane wrapping
Πα0 = RΠα . If as in [26] we consider that each (T2 )i has a rectangular geometry, a U (Na )
gauge group will arise from wrapping Na D6-branes on Πa and also on Πa0 , where
Πa
(n1a , m1a )
:
(n2a , m2a )
(n3a , m3a )
Πa0 : (n1a , −m1a ) (n2a , −m2a ) (n3a , −m3a )
(5.1)
In order to obtain a gauge group U (Na )×U (Nb ) we also need to place Nb D6-branes on Πb
and Πb0 . The spectrum of 4d left-handed chiral fermions in bifundamental representations
is then given by [26]
6
T
Iab
(Na , N̄b )
6
T
1
2
3
where Iab
0 = Iab0 Iab0 Iab0 =
Q3
i
i
i=1 (na mb
+
6
T
Iab
0 (Na , Nb )
(5.2)
+ nib mia ). In addition there are 4d chiral fermions
arising from the intersection of Πa with its orientifold image Πa0 , that transform in the
symmetric and antisymmetric representation of U (Na ), namely we have
a
1 T6
(I 0 − 8m1a m2a m3a )
2 aa
+
a
1 T6
(I 0 + 8m1a m2a m3a )
2 aa
(5.3)
The same orientifold projection can be performed for the toroidal orbifold T6 /Z2 × Z02 ,
also by modding out the theory by ΩR. Again, a rigid D6-brane wrapping Πa will have an
orientifold image wrapping Πa0 . It is easy to see that Πa and Πa0 will go through the same
fixed points on each (T2 )i , and so adding D6-brane orientifold images will not break the
discrete flavor symmetry of the model any further. In order to obtain the chiral spectrum
in this background one must consider (5.2) and (5.3) and project out all the chiral modes
that are not invariant under the orbifold action. Following our discussion of section 2.2,
it is easy to see that (5.2) is replaced by
Iab (Na , N̄b )
+
Iab0 (Na , Nb )
(5.4)
where Iab is given by (2.10) and similarly for Iab0 with the replacement b → b0 . The
orbifold projection of (5.3) is less straightforward but one can check that it amounts to
a
1
T6
(Iaa0 + 4 IaO6
)
2
+
a
1
T6
(Iaa0 − 4 IaO6
)
2
(5.5)
where for computing Iaa0 we use again the expression (2.10), but with the wrapping
6
T
numbers of Πa0 instead of Πb . On the other hand, IaO6
is the T6 intersection number (2.4)
33
between Πa and the three-cycle ΠO6 , with
[ΠO6 ] = −2 [a1 ] · [a2 ] · [a3 ] + [a1 ] · [b2 ] · [b3 ] + [b1 ] · [a2 ] · [b3 ] + [b1 ] · [b2 ] · [a3 ]
(5.6)
The three-cycle (5.6) has a geometrical interpretation, namely that the orientifold
projection ΩR introduces a set of O6-planes that are located at the fixed point loci of
ΩR, ΩRΘ, ΩRΘ0 and ΩRΘΘ0 , and adding up the homology classes of all these threecycles we can associate a total homology class [ΠO6 ] for the O6-plane. If each (T2 )i has a
rectangular geometry such homology class is given by (5.6). We refer the reader to [16] for
other cases in which some (T2 )i is not rectangular, and for a generalization of eqs.(5.4),
(5.5) to these cases.14
The importance of introducing O6-planes is that they allow to construct consistent and
supersymmetric D6-brane models [32]. In general, a D6-brane model will be consistent if
and only if the RR-tadpole condition
X
Nα ([ΠFα ] + [ΠFα0 ]) = 4[ΠO6 ]
(5.7)
α
is satisfied. Here the index α runs over each of the D6-branes of the model, ΠFα stands for
the fractional three-cycles described in appendix A and ΠFα0 is the image of ΠFα under R.
As discussed in appendix A we can describe a D6-brane on ΠFα in terms of T6 wrapping
numbers (niα , miα ). One can then see that the condition for a D6-brane model to preserve
the N = 1 supersymmetry of the T6 /Z2 × Z02 × ΩR background is [31]
θα1 + θα2 + θα3 = 0 mod 2π,
∀α
(5.8)
mi R
with θαi = tan−1 niαRxyi . As shown in [16], both conditions (5.6) and (5.8) are equivalent
α
i
to simple expressions in terms of the T6 wrapping numbers (niα , miα ) and can be satisfied
simultaneously. In the next subsection we will consider a set of D6-branes which are a
Pati-Salam subsector of a D6-brane model built in [16] satisfying both conditions.
14
Our conventions are such that a positive intersection number Iab signals a net amount of |Iab | 4d left-
handed chiral fermions in the representation (Na , N̄b ), while a negative intersection signals |Iab | fermions
in the same representation but with opposite chirality. In [16] this chirality convention is reversed.
34
5.2
A global Pati-Salam four-generation model
As an example of D-brane model with non-trivial flavor symmetry group let us consider
the intersecting D6-brane model in table 8 of [16], which is based on the orientifold
background T6 /Z2 × Z02 × ΩR. In particular, we will consider the subsector given by the
D6-branes a1 , a2 and a3 in that model, whose wrapping numbers we display in table 4.
(n1α , m1α ) (n2α , m2α ) (n3α , m3α )
Nα
Na1 = 4
(1, 0)
(0, 1)
(0, −1)
Na2 = 2
(1, 0)
(2, 1)
(4, −1)
Na3 = 2
(−3, 2)
(−2, 1)
(−4, 1)
Table 4: Wrapping numbers for the four-generation Pati-Salam model of [16].
The gauge group that arises from this set of D-branes is given by U (4) × U (2) ×
U (2), and as shown in [16] the chiral spectrum contains four families of left-handed chiral
fermions in the representations (4, 2, 1) + (4̄, 1, 2). We then have a four-generation PatiSalam model, and because the supersymmetry conditions (5.8) amount to impose
tan−1
2U
3
1
2U 2 = U 3
2
3
+ tan−1 U2 + tan−1 U4 = π
(5.9)
with U i = Ryi /Rxi , we can find a continuum of supersymmetric solutions. The matter
spectrum then contains 4(4, 2, 1) + 4(4̄, 1, 2) N = 1 left-handed chiral multiplets.
Flavor group and representations
Let us analyze this Pati-Salam model in light of the results of section 2. Table 5 shows
i
the toroidal intersection numbers Iαβ
for each of the relevant sectors of this model. Notice
that all these numbers are even, and so in (3.15) one has that d1 = d2 = d3 = 2. That is,
the flavor symmetry group of this sector is given by
(1)
(2)
(3)
2 a3
PaT16a/Z
× D4 × D4
0 = D4
2 ×Z
2
(5.10)
(i)
where the factor D4 arises from the symmetries on the two-torus (T2 )i . While (5.10)
corresponds to a symmetry of the Pati-Salam sector, it does not need to be respected by
35
Sector
U (4) × U (2)L × U (2)R
1
Iαβ
2
Iαβ
3
Iαβ
Projection
Iαβ
a1 a2
(4, 2̄, 1)
0
-2
4
−Ie2 Io3
2
a1 a02
(4, 2, 1)
0
-2
4
−Ie2 Io3
2
a1 a3
(4, 1, 2̄)
2
2
-4
Ie1 Ie2 Io3
-4
a1 a03
(4, 1, 2)
-2
2
-4
−
0
a2 a3
(1, 2, 2̄)
2
4
0
Ie1 Ie2
6
a2 a03
(1, 2, 2)
-2
0
-8
-Ie1 Ie3
-10
a2 a02
(1, 1+2 , 1)
0
-4
8
Io2 Ie3 − Ie2 Io3
-5+9
a3 a03
(1, 1, 1+2 ) &(1, 1, 3)
12
4
8
Ie1 Ie2 Ie3 + Io1 Io2 Io3
105+15
Table 5: Bulk intersection numbers together and the wavefunctions surviving the orbifold action in the four-generation Pati-Salam model. The last column shows the total
intersection number. A positive intersection number indicates a left-handed N = 1 chiral
multiplet, and a negative one a right-handed chiral multiplet.
the whole D-brane model. Indeed, in order to satisfy the consistency conditions (5.7) we
will need to add extra sets of D6-branes to those of table 5, and in order for (5.10) to be
i
an exact symmetry of the model all the intersection numbers Iαβ
involving these extra
D6-branes also need to be even. In general one would not expect this to be the case and,
indeed, by looking at the completion of this model given by table 8 of [16] one realizes
that there is always some D6-brane β of this extra set such that Iai j β = odd for any given
i. The flavor symmetry group (5.10) is then broken by the presence of the other D-branes
of this model, and can only be thought as an approximate symmetry of the Pati-Salam
sector of table 5. Nevertheless, even if not exact this symmetry will constrain the Yukawa
couplings of this model at tree-level and, because of N = 1 supersymmetry, at all orders
in perturbation theory. It is then useful to analyze under which representation of the
discrete flavor group (5.10) transform each of the chiral modes of table 5.
Let us for instance consider the sector a1 a2 of this model, which contains 2 copies
of left-handed multiplets in the representation (4, 2̄, 1). In order to see how these two
copies arise we must compute the combination of even or odd points on each two-torus
that survive the orbifold projection. Following our discussion of section 2 and due to
36
the particular signs of Iai 1 a2 for i = 1, 2, 3 one is instructed to keep the wavefunctions of
j2
j3
j3
j2
ψeven
, counted with the appropriate chirality. More precisely,
or ψodd
the form ψeven
ψodd
the chiral index in this sector is given by Io2 Ie3 − Ie2 Io3 = 0 − (−2) = 2, in agreement
with [16]. In fact, because Ia21 a2 = −2 there are no odd wavefunctions in (T2 )2 and so the
wavefunctions that correspond to this sector are of the form
j3
j2
· ψodd
ψaj21 a2 = ψeven
j2 = 0, 1,
j3 = 0
(5.11)
By looking at table 2 one can see how these chiral modes transform under the flavor group
Pa1 a2 a3 . On the one hand the index j2 transforms in the 2-dimensional representation R2
(2)
of D4 , and on the other hand the index j3 only takes one value and transforms as (−, −)
(3)
under the flavor subgroup D4 .
y3
y2
y1
3
1
2
1
x1
0
x2
0
x3
Figure 5: Branes a1 (red) and a2 (blue) with labels for the different intersection points.
Geometrically, one can understand this result by drawing both D6-branes and labelling
their intersection points as pj2 with j = 0, 1 in (T2 )2 and pk3 with k = 0, 1, 2, 3 in (T2 )3 ,
see Figure 5. Clearly, the two points in the second torus are even under the orbifold
action and there are no odd points. In the third torus we find three even points, namely,
p3 = {0, 1+3, 2} and an odd one given by p3 = {1−3}. Since the orbifold action selects the
points whose parity are (odd,even) or (even,odd), we have that the surviving points are
(p2 , p3 ) = ({0}, {1−3}) and (p2 , p3 ) = ({1}, {1−3}) which correspond to the two different
chiral modes in (5.11). One can now see how the translation z2 7→ z2 + τ /2 interchanges
these two points, while they pick up a minus sign under the translation z3 7→ z3 + τ /2.
Adding up the action of the discrete B-field transformation (see footnote 7) we indeed
(2)
(3)
recover that these two zero modes transform as R2 ⊗ (−, −) under D4 × D4 . Finally,
37
it is easy to see that the D6-brane intersections, which are the line {y1 = 0} in (T2 )1 , are
(1)
invariant under the translation z1 7→ z1 + 1/2 and in general by the full action of D4 .
The final result has been summarized in table 6, together with the representations for the
other sectors of the form ai aj and ai aj 0 with i 6= j, that can be treated similarly.
(1)
D4
(2)
(3)
Sector
Field
D4
D4
a1 a2
FL = (4, 2̄, 1)
1
R2
(−, −)
a1 a02
FL0 = (4, 2, 1)
1
(−, −)
R2
a1 a3
FR = (4̄, 1, 2)
R2
R2
(−, −)
a2 a3
H = (1, 2, 2̄)
R2
1 ⊕ (+, −) ⊕ (−, +)
1
a2 a03
H 0 = (1, 2̄, 2̄)
R2
1
12 ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
Table 6: Representations of the Pati-Salam fields under the flavor symmetry group.
Yukawa couplings
Given the above representations under the flavor symmetry group one can now consider
the Yukawa couplings
Y : (a1 a2 ) ⊗ (a1 a3 ) ⊗ (a2 a3 ) −→ (4, 2̄, 1) ⊗ (4̄, 1, 2) ⊗ (1, 2, 2̄)
Y 0 : (a01 a2 ) ⊗ (a1 a3 ) ⊗ (a02 a3 ) −→ (4, 2, 1) ⊗ (4̄, 1, 2) ⊗ (1, 2̄, 2̄)
(5.12)
which are allowed by gauge invariance.15 It however happens that several of these couplings are not allowed by the discrete flavor symmetry (5.10), as we will now see.
Let us first consider the coupling Y in (5.12). In principle, Y has 2×4×6 independent
components since there are 6 different Higgses that appear in this set of Yukawa couplings.
Nevertheless, in general the discrete symmetries in each torus will reduce the number of
(1)
independent Yukawas. On the one hand, invariance under D4
forces us to choose the
singlet in 1 ⊗ R2 ⊗ R2 = 1 ⊕ (+, −) ⊕ (−, +) ⊕ (−, −) which reduces by a factor 4 the
(2)
number of independent Yukawas. On the other hand, under D4 the coupling Y behaves
15
This includes those Abelian discrete gauge symmetries that remain after the U (1) factors of the gauge
group are broken by a Stückelberg mechanism [7].
38
as follows
R2 ⊗ R2 ⊗ (1 ⊕ (+, −) ⊕ (−, +)) = (1 ⊕ (+, −) ⊕ (−, +) ⊕ (−.−)) ⊗ (1 ⊕ (+, −) ⊕ (−, +))
= 13 ⊕ (+, −)3 ⊕ (−, +)3 ⊕ (−, −)3
which reduces by another factor of 4 the number of independent Yukawas. Finally, since
(3)
D4
does not impose further constraints we conclude that there are only
2×4×6
4×4
= 3
independent components in Y . In other words, at tree-level there will only be three
independent Yukawas within this sector. More precisely one obtains the following Yukawa
couplings Yijk FL,i FR,j Hk where


aH0 + cH2
bH1
aH3 + cH5
bH4

Yijk Hk = 
bH1
aH0 − cH2
bH4
aH3 − cH5
(5.13)
where the row index i runs over the two families of left-handed multiplets FL in the
a1 a2 sector, while the index j runs over the four families of right-handed multiplets FR .
For concreteness we have displayed the definition of these multiplets in terms of D-brane
intersections in table 7.
FL,i
FR,j
Hk
(ψ 0 )2 · (ψ 1 − ψ 3 )3
(ψ 0 )1 · (ψ 0 )2 · (ψ 1 − ψ 3 )3
(ψ 0 )1 · (ψ 0 + ψ 2 )2
(ψ 1 )2 · (ψ 1 − ψ 3 )3
(ψ 0 )1 · (ψ 1 )2 · (ψ 1 − ψ 3 )3
(ψ 0 )1 · (ψ 0 − ψ 2 )2
(ψ 1 )1 · (ψ 0 )2 · (ψ 1 − ψ 3 )3
(ψ 0 )1 · (ψ 1 + ψ 3 )2
(ψ 1 )1 · (ψ 1 )2 · (ψ 1 − ψ 3 )3
(ψ 1 )1 · (ψ 0 + ψ 2 )2
(ψ 1 )1 · (ψ 0 − ψ 2 )2
(ψ 1 )1 · (ψ 1 + ψ 3 )2
Table 7: Wavefunctions of the fields in the Yukawa couplings (5.13). Here (ψ j )i stands
for a delta-function localized at the j th intersection of the D6-branes a1 and a2 in (T2 )i .
Considering now the Yukawa couplings Y 0 in (5.12), one finds that the effect of the
discrete flavor symmetry is even more dramatic since there is no combination which is
(2)
invariant under the factor D4 . As a result these Yukawa couplings will vanish and (5.13)
will be the only set of Yukawas at the perturbative level. Hence, this four-generation
39
Pati-Salam model will in fact have two families whose mass is generated perturbatively.
It would be interesting to see how non-perturbative effects can generate the Yukawa
couplings for the remaining two generations.
Mass terms and net chirality
Besides Yukawa couplings, discrete flavor symmetries may forbid other kinds of couplings
like mass terms between vector-like pairs of of zero modes. In the model at hand such
kind of pairs arise in the sector a2 a02 , whose total intersection number is given by Ia2 a02 = 4.
This signals that we have a net chirality of four left-handed chiral multiplets in the
representation (1, 1+2 , 1), where 1+2 stands for an antisymmetric representation of U (2).16
However, this net chirality does not signal the actual content of open string zero modes
of this sector. A careful analysis using the rules of subsection 2.2 shows that in fact there
are nine left-handed chiral multiplets (the ones arising from the wavefunctions of the form
(even,odd)) and five right-handed chiral multiplets (the ones from the sector (odd,even))
in the representation (1, 1+2 , 1).
Typically, one would not worry about this mismatch between the zero mode content
and the net chiral index, because the ten extra zero modes not accounted by Ia2 a02 naturally
arrange into five vector-like pairs that form singlets under the gauge group U (4)×U (2)L ×
U (2)R . Hence, one expects that the presence of loop corrections or extra compactification
ingredients like background fluxes will generate a mass term for these pairs not protected
by gauge invariance.
Nevertheless given a flavor symmetry one needs to check that these pairs of opposite
chirality zero modes also form singlets under the discrete flavor group. For the model
at hand, table 8 shows the charges of the different points (or wavefunctions) under the
flavor group (5.10) for the sector a2 a02 . From there one can see that one cannot form a
(2)
vector-like pair that is a singlet under the factor D4 . As a result, a mass term for any
vector-like pair is forbidden by the discrete flavor symmetry. Even if in this particular
case the flavor symmetry is approximate, the effect generating such mass term must also
16
More precisely, one computes the spectrum of this sector by applying eqs.(5.5), with Ia2 O6 = 4.
Hence one obtains a net number of four chiral multiplets in the antisymmetric of U (2)L and no matter
in the symmetric representation of U (2)L .
40
Sector
(1)
(2)
D4
(3)
D4
D4
(even,odd)
1
1 ⊕ (+, −) ⊕ (−, +)
(+, −) ⊕ (−, +) ⊕ (−, −)
(odd,even)
1
(−, −)
12 ⊕ (+, −) ⊕ (−, +) ⊕ (−, −)
Table 8: Representations under the dihedral groups of the zero modes in a2 a02 .
break the flavor symmetry (like e.g. non-perturbative effects), and so we expect that such
masses for vector-like pairs are smaller that the ones allowed by all sort of symmetries.
The center of the flavor group
While the flavor symmetry group (5.10) is non-Abelian, its center can be compared with
other discrete Abelian groups present in this model. In particular it can be compared
with the ZN discrete gauge symmetries contained in the U(1) factors of U (4) × U (2)L ×
U (2)R . These discrete gauge symmetries are discussed in appendix C following the general
prescription of [7]. The result is that they are trivial in the sense that they reduce to the
center of the gauge group SU (4) × SU (2)L × SU (2)R , which is generated by the elements
g4 = diag(i, i, i, i),
g2,L = diag(−1, −1)L ,
g2,R = diag(−1, −1)R .
(1)
(5.14)
(2)
(3)
Let us denote the center of the gauge group by Z(G) and the center of D4 ×D4 ×D4
by Z(P ). We would like to know if any subgroup of Z(P ) is contained in Z(G) when
acting on the Pati-Salam model. Both groups are finite so they have a finite collection of
subgroups and this can be answered by direct computation. Table 9 shows the charges of
the visible sector under every Z2 subgroup of Z(P ) and Z(G).
(1)
Looking at Table 9 we see that Z2 and Z2,R are the same. Also, the Z2 generated
(3)
(2)
by the product of the generators of Z2 and Z2,C is equivalent to Z2 which shows that
(2)
(3)
Z2 and Z2 are not independent but are related by a gauge transformation. We thus
(1)
(2)
(3)
(1)
(3)
find the discrete flavor group is actually D4 × D4 × D4 /(Z2 × Z2 ), and that all the
couplings forbidden by this symmetry should be understood in terms of this quotient.
41
(1)
(2)
(3)
Sector
Z2
Z2
Z2
Z2,C
Z2,L
Z2,R
a1 a2
+
−
+
−
−
+
a1 a3
−
−
+
−
+
−
a2 a3
−
+
+
+
−
−
a1 a02
+
+
−
−
−
+
a2 a03
−
+
+
+
−
−
a2 a02
+
+
+
+
+
+
a3 a03
+
+
+
+
+
+
Table 9: Charges of the visible sector under Z2 subgroups of Z(P ) and Z(G).
5.3
A local Pati-Salam three-generation model
Besides the four-generation model of [16], one may construct other models in Z2 × Z02 with
semi-realistic spectrum that also display a non-trivial discrete flavor symmetry. In the
following we analyze a simple three-generation Pati-Salam model where families transform
with non-Abelian representations under a Dihedral flavor group.
The D6-brane content of the model is shown in table 10, where the wrapping numbers
n, l are arbitrary positive integers. Again, this D6-brane content is not sufficient to satisfy
Nα
(n1α , m1α )
(n2α , m2α )
(n3α , m3α )
Na = 4
(1, 0)
(1, 1)
(1, -1)
Nb = 2
(n, -3)
(0, 1)
(3, -1)
Nc = 2
(l, -1)
(-2, 1)
(-1, -1)
Table 10: Wrapping numbers for the three-generation Pati-Salam model.
the RR-tadpole conditions (5.7), and extra D-branes should be added in order to construct
a complete model. We will then consider it as a local Z2 × Z02 model, whose discrete flavor
symmetry may or may not be broken by the extra D-branes that complete it.
It is easy to see that the gauge group that arises from this D6-brane content is again
given by U (4) × U (2)L × U (2)R , and that now the supersymmetry conditions amount to
42
U2 = U3
−1 3U 1
−1 U 3
tan
+ tan
= π2
n
3
2
1
tan−1 Ul + tan−1 U2 − tan−1 U 3 = 0
i
(5.15)
1
where U = Ryi /Rxi . One can solve these equations by setting n > l > 0, U =
q
2n
and U 2 = U 3 = n−l
, hence finding again a N = 1 Pati-Salam model.
q
n(n−l)
2
The chiral spectrum of this model can be found by computing the intersection numbers
on each two-torus and applying the results of subsection 2.2. The result is displayed in
table 11, from where it is manifest that all the intersection points in the third torus are
even. We then conclude there is a flavor symmetry group of the form
Pabc
T6 /Z2 ×Z02 = D4
(5.16)
where D4 is generated by translations and B-field transformations on (T2 )3 . The zero
mode spectrum in the bc sector depends on the integer wrapping numbers n > l > 0. In
the table we have considered the choice n = 2, l = 1, which gives a minimal Higgs sector.
Sector
U (4) × U (2)L × U (2)R
1
Iαβ
2
Iαβ
3
Iαβ
Projection
Iαβ
ab
(4̄, 2, 1)
-3
1
2
Io1 Ie2 Ie3
-2
ab0
(4̄, 2̄, 1)
3
-1
4
Io1 Ie2 Io3
-1
ac
(4, 1, 2̄)
-1
3
-2
Ie1 Io2 Ie3
2
ac0
(4, 1, 2)
-1
1
0
Ie1 Ie2
1
bc
(1, 2̄, 2)
1
2
-4
Ie1 Ie2 Io3
-2
bc0
(1, 2, 2)
5
2
2
Ie1 Ie2 Ie3
12
bb0
(1, 1+2 , 1)
12
0
6
Ie1 Ie3
18
Table 11: Bulk intersection numbers of the model of table 10 with n = 2 and l = 1,
together with the points surviving the orbifold action and the total intersection number.
Similarly to the previous example we can easily extract the representation of these
chiral Pati-Salam families under the flavor symmetry group D4 . We present the result of
43
this analysis in table 12, which shows that in this model one generation is different in the
sense that it transforms under an Abelian representation of D4 , while the other two form
a doublet of the fundamental representation R2 of the Dihedral group.
Sector
Fields
D4
ab
FR = (4̄, 2, 1)
R2
ab0
FR0 = (4̄, 2̄, 1)
(−, −)
ac
FL = (4, 1, 2̄)
R2
ac0
FL0 = (4, 1, 2)
(+, +)
bc
H = (1, 2̄, 2)
(−, −) ⊕ (−, −)
bc0
H 0 = (1, 2, 2)
⊕ R2
6
Table 12: D4 representations.
The only Yukawas allowed by gauge invariance (including anomalous U (1)’s) are
Y : ab ⊗ ac ⊗ bc −→ (4̄, 2, 1) ⊗ (4, 1, 2̄) ⊗ (1, 2̄, 2)
(5.17)
Y 0 : ab0 ⊗ ac ⊗ bc0 −→ (4̄, 2̄, 1) ⊗ (4, 1, 2̄) ⊗ (1, 2, 2).
(5.18)
and under the discrete D4 these coupling behave as
Y
Y0
: R2 ⊗ R2 ⊗ [(−, −) ⊕ (−, −)] = 1 ⊕ 1 ⊕ . . .
6
: (−, −) ⊗ R2 ⊗ ⊕6 R2 = ⊕ 1 ⊕ . . .
(5.19)
(5.20)
where the dots stand for nontrivial representations of D4 , and we used R2 ⊗ R2 = 1 ⊕
(+, −) ⊕ (−, +) ⊕ (−, −). We then conclude that there are a total of eight independent
parameters in the Yukawa couplings given by Y and Y 0 .
6
Conclusions and outlook
In this paper we have analyzed the presence of discrete flavor symmetries in models of
intersecting and magnetized D-branes. The general principle to determine the flavor symmetry is to first consider the group Pbulk of non-trivial metric and B-field transformations
that leave the closed string background invariant. In the absence of D-branes, this group
44
of transformations is part of the gauge symmetry group of the 4d effective theory. In
the presence of D-branes this gauge group will be partially broken, because the D-brane
background is not invariant under its action. The subgroup of isometries and B-field
transformations that leave both the closed and open string backgrounds invariant will
generate a group of discrete flavor symmetries which in general will be non-Abelian.
We have implemented the above principle in compactification manifolds like T6 and
(T2 )3 /Z2 × Z02 and orientifolds thereof. In the case of T6 the initial group of bulk gauge
symmetries arising from the metric and B-field is continuous, namely Pbulk = U (1)12 , and
so one can apply the techniques of [8] to obtain via dimensional reduction the symmetry
group that remains after D-branes have been introduced. The result is a discrete flavor
group that fully agrees with the definition in the previous paragraph, as we have checked
for models of intersecting and magnetized D-branes. In the case of the Z2 × Z02 orbifold
the bulk gauge group is discrete, namely Pbulk = Z12
2 and so we cannot apply the approach
of [8].17 However, the strategy followed in this paper does still apply, and so we are able
to compute the flavor symmetry group also for this case. The same works for orientifold
quotients of the above backgrounds, which in turn allows to study the flavor symmetries
of consistent, 4d N = 1 chiral D-brane models like the ones constructed in [16]. We have
then analyzed the flavor symmetry group of a couple of semi-realistic Pati-Salam examples,
obtaining that the matrices of Yukawa couplings are constrained by the flavor symmetry
group beyond the already well-known effect of massive U(1) D-brane symmetries and the
Abelian discrete gauge symmetries contained in them [7].
One of the most attractive features of this approach is its generality, which allows
to extend our results in a number of ways. While we have focused on the factorizable
orbifold (T2 )3 /Z2 × Z02 one can easily generalize our observations to D6-brane models on
non-factorizable T6 /Z2 × Z02 geometries [25], or to other orbifold geometries like T6 /ZN
or T6 /ZN × ZM where more realistic D-brane models have been constructed (see e.g.
[47–50]). It would be interesting to see which flavor symmetry groups arise in these other
orbifold backgrounds, and in D6-brane models based on smooth Calabi-Yau geometries
17
Using the approach of [10] one shoud be able to embed this discrete bulk gauge group into a continuous
one. It would be interesting to also include the presence of D-branes into the formalism of [10] in order
to have an alternative derivation of the flavor symmetry group in manifolds with discrete isometries.
45
with discrete isometries [51–53]. Also while we have considered D-branes that either
intersect or carry worldvolume fluxes, one may extend this approach to D-branes that
both intersect and are magnetized, like type IIA models with coisotropic D8-branes [54] or
type IIB models based on D7-branes (see e.g. [55, 56]). Finally, having a CFT description
of the closed string background is not essential in this approach, so one may also extend it
to, e.g., type I compactifications with both open and closed string background fluxes [57].
Another interesting consequence of this approach is that it provides a useful notion of
approximate flavor symmetries. Here the principle is again quite simple. If a subgroup
Pabc of the bulk gauge symmetry group Pbulk leaves a subset A = {a,b,c} of three background D-branes invariant, then the sector of the theory given by U (Na ) × U (Nb ) × U (Nc )
will respect the symmetry Pabc , and in particular the tree-level couplings between open
string modes of this sector will be invariant under it. If now there is a fourth D-brane d
which is not invariant under Pabc this flavor symmetry group will be broken, and can only
be thought as an approximate symmetry of the gauge sector U (Na ) × U (Nb ) × U (Nc ).
Nevertheless, in supersymmetric models the holomorphic Yukawa couplings that arise
from the subset A of D-branes will be constrained by the discrete symmetry Pabc at all
orders in perturbation theory. Hence, the holomorphic Yukawas forbidden by Pabc will
only be generated at the non-perturbative level, being thus naturally suppressed with
respect to the allowed ones. It would be interesting to explore such scenario in specific
D-brane models like the ones considered here, in a similar spirit to [58–60].
Acknowledgments
We would like to thank P. G. Cámara, C. Hagedorn, L. E. Ibáñez and A. M. Uranga for
useful discussions. This work has been partially supported by the grants FPA2009-07908
and FPA2012-32828 from MINECO, HEPHACOS-S2009/ESP1473 from C.A. de Madrid,
the REA grant agreement PCIG10-GA-2011-304023 from the People Programme of FP7
(Marie Curie Action), the grant SEV-2012-0249 of the Centro de Excelencia Severo Ochoa
Programme and the SPLE Advanced Grant under contract ERC-2012-ADG 20120216320421. F.M. is supported by the Ramón y Cajal programme through the grant RYC2009-05096. D.R. is supported through the FPU grant AP2010-5687. L.V. is grateful for
support from CONACyT and to the IFT-UAM/CSIC for hospitality.
46
A
The Z2 × Z02 orbifold
Let us consider type IIA string theory compactified on the toroidal orbifold background
T6 /(Z2 × Z2 ), the Z2 generators acting as


 z1 → −z1
Θ:
z2 → −z2

 z →z
3
3


 z1 → z1
0
Θ :
z2 → −z2

 z → −z
3
3
(A.1)
on the three complex coordinates of T6 = (T2 )1 × (T2 )2 × (T2 )3 . Besides such action one
must specify the choice of discrete torsion that relates these two Z2 group generators. As
explained in [46] there are two inequivalent choices, whose twisted homologies are either
tw.
tw.
tw.
(htw.
11 , h21 ) = (48, 0) or (h11 , h21 ) = (0, 48). Similarly to [16] we will consider the second
case, and dub it as Z2 × Z2 orbifold with discrete torsion or Z2 × Z02 . Such background
then contains 96 collapsed three-cycles at the fixed loci of (A.1).
Let us now add space-time filling D6-branes wrapping supersymmetric three-cycles of
this toroidal orbifold. In terms of a factorized T6 geometry these can be described as the
product of three one-cycles
[Πa ] =
3
O
nia [ai ] + mia [bi ]
nia , mia ∈ Z and coprime
(A.2)
i=1
i
i
where [a ], [b ] are the homology classes of the fundamental one-cycles of (T2 )i . Notice
that the T6 homology class [Πa ] is invariant under the orbifold action (A.1), and so one can
consider three-cycle representatives Πa also invariant under (A.1). A D6-brane wrapping
an invariant three-cycle will suffer the orbifold projection on its Chan-Paton degrees of
freedom, resulting into fractional D6-branes with non-vanishing charge under the RR
twisted sector of the theory. Geometrically, on each (T2 )i a fractional D6-branes goes
through two fixed points of the action zi → −zi , and it wraps collapsed three-cycles that
correspond to such fixed points (see fig. 6). Precisely because of this, fractional D6-branes
are ‘rigid’: they cannot be taken away from a fixed locus of the action (A.1), and so they
do not contain the deformation moduli typical of D-branes in toroidal compactifications.
Following [16] the homology class of a fractional D6-brane is of the form






X
X
1
1 X Θ
0
0
Θ0
ΘΘ0 
 1
 1
a,IJ [ΠΘ
Θ
ΘΘ
[ΠFa ] = [ΠB
a ]+
IJ, a ] +
a,JK [ΠJK, a ] +
a,IK [ΠIK, a ]
4
4 I,J∈S a
4 J,K∈S a
4 I,K∈S a
Θ0
Θ
ΘΘ0
(A.3)
47
6
where [ΠB
a ] stands for a bulk three-cycle, that is a T three-cycle of the form (A.2) that
is inherited in the orbifold quotient. Bulk three-cycles correspond to the untwisted RR
charges of the orbifold, and the intersection number between them is given by
B
Iab
=
[ΠB
a]
·
[ΠB
b ]
3
Y
= 4 (nia mib − mia nib ),
(A.4)
i=1
where the factor of 4 arises from taking into account the Z2 × Z02 orbifold action. Beside
the bulk cycles there are 32 collapsed three-cycles for each of the three twisted sectors.
Their homology class is of the form
[ΠgIJ, a ] = 2[egIJ ] ⊗ niag [aig ] + miag [big ]
(A.5)
where g = Θ, Θ0 , ΘΘ0 runs over all twisted sectors, and ig = 3, 1, 2, respectively. Here egIJ
I, J ∈ {1, 2, 3, 4} stand for the 16 fixed points on (T2 )i × (T2 )j /Z2 , where Z2 = {1, g} and
(T2 )i,j are the two-tori such that g : (zi , zj ) 7→ (−zi , −zj ) (see fig. 6). These fixed points
correspond to the Z2 singularities of a K3 surface in its orbifold limit T4 /Z22 , and each
can be blown up to a P1 whose homology class is given by [egIJ ]. Finally, aig , big stand
for the fundamental one-cycles of (T2 )ig , the two-torus which is left invariant under the
action of g. Gathering all these facts together, one can compute the intersection number
of two collapsed three-cycles as
i
i
[ΠgIJ, a ] · [ΠhKL, b ] = 4 δIK δJL δ gh (niag mbg − miag nbg )
y3
y2
y1
2
4
1
3
x1
(A.6)
2
4
1
3
x2
2
4
1
3
Figure 6: Fractional brane passing through 4 fixed points for each twisted sector. Fixed
points are denoted by dots in the Θ sector, by circles in the Θ0 sector and squares in the
ΘΘ0 sector.
48
x3
The homology class (A.3) is given by a particular linear combination of bulk and
collapsed three-cycles, which is determined as follows. In the covering space (T2 )3 a
BPS D6-brane looks like as a product of three 1-cycles with wrapping numbers (nia , mia )
and constant slope, see figure 6. Fractional D6-branes must be invariant under (A.1),
and so on each two-torus they must pass through two fixed points of (T2 )i /Z2 with
Z2 = {1, zi 7→ −zi }. Which are these fixed points depends on the wrapping numbers
(nia , mia ), see table 1 in the main text.
Let us now consider a particular twisted sector, say g = Θ. The collapsed three-cycles
2
2
of this sector are related to the fixed points eΘ
IJ of (T )1 × (T )2 /{1, Θ}. A fractional
D6-brane will pass through 4 fixed points eΘ
IJ . More precisely, the index I will take two
different values specified by (n1a , m1a ) and one of the choices in table 1, while J will be
constrained by (n2a , m2a ). This subset of 2 × 2 elements {(I, J)} ⊂ {1, 2, 3, 4} × {1, 2, 3, 4}
a
is denoted as SΘa in (A.3), and similar definitions apply to SΘa 0 and SΘΘ
0 . It is easy to see
that given the bulk wrapping numbers (nia , mia ), i = 1, 2, 3 there are eight different choices
a
2 3
for specifying SΘa , SΘa 0 and SΘΘ
0 . From the viewpoint of the covering space (T ) , these
choices correspond to the 23 different locations that an invariant three-cycle can have.
0
0
Θ
ΘΘ
Besides Sga one needs to specify the signs Θ
a,IJ , a,JK , a,IK = ±1 that appear in (A.3).
These signs are not arbitrary but must fulfill several consistency conditions discussed
in [16]. One finds that there are essentially 8 inequivalent choices for these signs. In
general the set of fixed points is given by
SΘ = {{I1 , I2 } × {J1 , J2 }}
(A.7)
SΘ0 = {{J1 J2 } × {K1 K2 }}
SΘΘ0 = {{K1 K2 } × {I1 I2 }}
where iα , jα , kα , α = 1, 2 represent fixed point coordinates in the first, second and third
0
0
Θ
ΘΘ
T2 factors, respectively. If we fix Θ
I1 J1 = J1 K1 = K1 I1 = +1 then all the other ’s
0
ΘΘ
depend on only three independent signs. More precisely we have that Θ
I2 J1 = K1 I2 = I ,
0
0
0
0
0
Θ
Θ
ΘΘ
Θ
Θ
ΘΘ
Θ
I1 J2 = J2 K1 = J , J1 K2 = K2 I1 = K and I2 J2 = I J , J2 K2 = J K , K2 I2 = K I , with
I , J , K = ±1. The choice of these three signs can be interpreted as the choice of discrete
Wilson lines for a fractional D6-brane along each one-cycle.
g
Having fixed Sga and a,IJ
as above, there are four inequivalent choices of wrapping
numbers (nIa , mIa ) which correspond to the same bulk three-cycle ΠB
a but to different
49
fractional three-cycle ΠFa . These are given by
(n1a , m1a )
(n2a , m2a )
(−n1a , −m1a ) (−n2a , −m2a )
(n1a , m1a )
(−n1a , −m1a )
(n3a , m3a )
(n3a , m3a )
(A.8)
(−n2a , −m2a ) (−n3a , −m3a )
(n2a , m2a )
(−n3a , −m3a )
and can be interpreted as the four different Z2 × Z02 twisted charges that a fractional
D6-brane can have. Indeed, it is easy to see that one obtains a pure bulk D6-brane by
adding these four fractional D6-branes. One can further support this claim by computing
the chiral spectrum between two fractional D6-branes, as we now proceed to show.
Chiral index
Given two stacks of fractional D6-branes wrapped on ΠFa and ΠFb one can easily compute
the chiral spectrum of open strings with one endpoint on each of them. Indeed, let
us consider Na D6-branes wrapped on ΠFa and Nb D6-branes on ΠFb . Then the chiral
spectrum will be given by Iab left-handed chiral multiplets in the bifundamental (Na , N̄b )
representation of SU (Na ) × SU (Nb ). Here Iab = [ΠFa ] · [ΠFb ] is the topological intersection
number of the two three-cycles, and can be computed from (A.4), (A.6) and the fact that
an intersection number between a bulk and a collapsed three-cycle vanishes.
For instance, let us consider the case where ΠFa , ΠFb are such that they have trivial
g
discrete Wilson lines (a,b
= 1 in (A.3)) and they both intersect the origin of (T2 )3
(I1 = J1 = K1 = 1 in (A.7) for Sga,b ). Then the intersection number Iab is specified by the
bulk wrapping numbers
ΠFa : (n1a , m1a ) (n2a , m2a ) (n3a , m3a )
ΠFb : (n1b , m1b ) (n2b , m2b ) (n3b , m3b )
(A.9)
More precisely we find that
Iab = [ΠFa ] · [ΠFb ] =
1 1 2 3
1
2
3
ρ1 ρ2
ρ2 ρ3 + Iab
ρ1 ρ3 + Iab
Iab Iab Iab + Iab
4
(A.10)
i
where Iab
= nia mib − nib mia and ρi is defined as in (2.8). Despite the factor of 1/4 one
can check that such intersection number is always an integer, as required by consistency.
B
Notice that the bulk intersection number Iab
remains unchanged if we replace ΠFa by
50
i
any of the other bulk wrapping numbers in (A.8). The intersection numbers Iab
for each
individual two-torus do however depend on this choice, and so does the total intersection
number Iab . As show in the main text this formula is reproduced by considering those
linear combinations of intersection points invariant under the orbifold action. As one can
also see from that discussion, the four different type of projection that depend on the signs
s1 , s2 , s3 can be obtained by considering the pair of D6-branes (A.9) and then replacing
ΠFa by any of the other bulk wrapping numbers in (A.8). This shows in more detail that
each of these D6-branes has a different Chan-Paton factor, because for the same bulk
embedding the open strings ending in Πa feel a different orbifold action, adding up to the
regular representation of Z2 × Z02 .
B
Flavor symmetries from dimensional reduction
Let us consider the dimensional reduction that yields the Stückelberg lagrangians (3.1-3.6)
in 4d for D6-branes at angles in type IIA on T2 × T2 × T2 which shows the appearance
of discrete symmetries. In [8] this was done for a toroidal compactification of Type I with
magnetic fluxes. In our case we should consider the Type IIA supergravity together with
the DBI action for the branes at angles to get the full non-Abelian structure. We will,
however, take a simpler approach and consider only the DBI part to derive the abelian
part of the symmetry.
Consider a D6-brane wrapping a factorizable three-cycle Πa = (n1a , m1a ) ⊗ (n2a , m2a ) ⊗
(n3a , m3a ). The DBI action for such a D6-brane is
Z
Z
p
4
S6 = −µ6
dx
d3 q e−Φ − det(P [G] + P [B] − kF )
M4
(B.1)
Πa
with k = 2πα0 and P [·] is the pullback on the worldvolume of the brane which looks like
P [A]αβ = Aαβ + Aij ∂α φi ∂β φj + ∂α φi Aiβ + ∂β φi Aαi
(B.2)
where α, β are indices on the brane and i, j are transverse. φi are the embedding functions
of the brane in the bulk. Using the Taylor expansion of the determinant
1
1
det(1 + M ) = 1 + Tr M + [Tr M ]2 − Tr M 2 + . . .
2
2
51
(B.3)
we can expand the action (B.1) in derivatives. Namely,
Z
Z
p
1
4
3
−Φ0
S6 = −µ6
dx
d qe
− det Gαβ 1 + Gαβ Gij ∂α φi ∂β φj + Gαβ ∂α φi Giβ
2
M4
Πa
k
k2
αβ
αβ
− Bαβ F + Fαβ F + . . .
(B.4)
2
4
where we only kept the terms quadratic in fluctuations. Since the brane is wrapping the
cycle Πa we take the following rotated coordinates in T2 × T2 × T2
q l = xl cos θl + y l sin θl ,
pl = −xl sin θl + y l cos θl ,
tan θl =
mla
nla
(B.5)
with xl , y l real coordinates on (T2 )l for l = 1, 2, 3, the q l ’s are along the brane and the
pl ’s transverse to it. Going back to the action (B.4) we get the following terms
Z
Z
p
1 µν
3
−Φ0
4
− det Gαβ
G Gpp ∂µ φp ∂ν φp + Gµν ∂µ φp Gpν
S6 ⊃ −µ6
d qe
dx
2
M4
Πa
k2
µq
µq
− kBµq F + Fµq F + . . . . (B.6)
2
In this expression the indices µ, ν are in 4d, while p and q run through pl and q l respectively.
The first line yields18
3
LSt
where we defined φia =
p
2
1X
= −
∂µ φia − mia Vµxi + nia Vµyi
2 i=1
(B.7)
n2i + m2i φi so that φia ∼ φia + 1 following the conventions in [8].
This is the Lagrangian (3.1) that describes the spontaneous breaking of the continuous
isometry group U (1)6 of the torus to U (1)3 × Zq1 × Zq2 × Zq3 with qi = (nia )2 + (mia )2 due
to the presence of the brane.19 Also, φia provide the longitudinal degree of freedom to the
massive gauge bosons −mia Vµxi + nia Vµyi .
Furthermore, from the second line in (B.6) one finds the following contribution to the
low energy action
3
LSt
where ξai =
18
p
2
1X
∂µ ξai − nia Bµxi − mia Bµyi
= −
2 i=1
(B.8)
n2i + m2i Ai and we have rescaled the B-field as B → k −1 B, c.f.(3.6).
The kinetic term of the gauge bosons Vµx and Vµy that complete the Stückelberg Lagrangian can be
obtained from dimensional reduction of the closed string sector of the theory.
19
See section 2.5 in [9] for a discussion on the discrete part of this group.
52
This analysis shows that the presence of a single brane breaks the continuous U (1)12
gauge symmetry that arises from the reduction of the metric and B-field down to U (1)6
plus some discrete part.20 It is clear that adding more branes will generically break the
gauge symmetry completely. Indeed, the Lagrangian for a set of intersecting branes will
include the terms
3
LSt
2
2 o
1 XXn
= −
∂µ φiα − miα Vµxi + niα Vµyi + ∂µ ξαi − niα Bµxi − miα Bµyi
2 α i=1
(B.9)
where α runs over the branes. Unless all the branes are parallel in a given torus this will
Higgs the continuous part of the gauge group completely. Nevertheless, there can be a
discrete remnant which we discuss in the following.
Let us restrict to the case where there are only two branes a and b and focus on the
part of the action (B.9) that involves Vµxi , Vµyi . Following [8], one can see that the discrete
gauge group coming from the i-th torus is
Tiab =
Γi
Γ̂i
(B.10)
where Γi is the lattice of the i-th torus and Γ̂i is the lattice generated by the intersection
points. Namely,
Γi = h(1, 0), (0, 1)i,
Γ̂i =
1
h(nia , mia ), (nib , mib )i.
i
Iab
(B.11)
i and since the three-cycles the D6-branes wrap are
One can check that indeed Tiab = ZIab
factorizable we have
1 × ZI 2 × ZI 3
TTab6 = ZIab
ab
ab
(B.12)
which reproduces eq.(3.3) in the main text. A completely analogous argument shows that
the second term in (B.9) yields
1 × ZI 2 × ZI 3
WTab6 = ZIab
ab
ab
(B.13)
in agreement with eq.(3.8).
20
In case that we have an orientifold background the bulk symmetry is not U (1)12 but U (1)6 × Z62 .
Indeed, because the O6-planes are located along y i = 0, 21 for i = 1, 2, 3, the U(1) symmetries generated
by Vµyi and Bµxi are broken down to Z2 , obtaining a residual Z2 × Z2 gauge group for each (T2 )i .
53
These two groups, TTab6 , WTab6 , do not commute as can be seen from their action on the
wavefunctions of chiral matter in the ab sector. Instead they generate the non-Abelian
discrete group
1 × HI 2 × HI 3
Pab
T6 = HIab
ab
ab
(B.14)
with HN ' (ZN × ZN ) o ZN .
Magnetized D-branes
In order to connect with the dimensional reduction of Type I with magnetized D9-branes
performed in section 6.2 of [8] we T-dualize the above setup along the directions y i for
i = 1, 2, 3. Thus, as usual D6 branes at angles turn into D9 branes with magnetic fluxes
given by
Fxai yi =
mia
Ini .
knia a
(B.15)
Notice that since the three-cycles the D6-branes wrap are factorizable the ‘nondiagonal’
fluxes Fzi z̄j are zero for i 6= j. This means that just like before the dimensional reduction
on T6 factorizes in (T2 )1 × (T2 )2 × (T2 )3 . More precisely, the above computation gives
in this case
3
LSt
o
1 XXn
i
i
xi
i
yi 2
i
i
xi
i
yi 2
= −
∂µ ξx,α − nα Bµ − mα Vµ
+ ∂µ ξy,α + mα Vµ − nα Bµ
(B.16)
2 α i=1
i
i
correspond to the Wilson lines on the worldvolume of the brane
, ξy,α
where the axions ξx,α
i
i
+ 1. In order to
∼ ξq,α
along xi and y i respectively and have periodic identifications ξq,α
cancel the total D9-charge we must include an orientifold projection acting trivially on
the tori which introduces O9-planes. Also, the B-field does not survive the projection so
we can safely set it to zero21 in (B.16) which reproduces the result in [8], namely,
(
2 2 )
3
i
i
m
m
1 XX
i
i
LSt = −
∂µ ξx,α
− iα Vµyi + ∂µ ξy,α
+ iα Vµxi
(B.17)
2 α i=1
nα
nα
i
i
with ξq,α
∼ ξq,α
+ 2/niα .22
21
Actually, a subgroup Z62 survives the orientifold projection, as in the type IIA case (see footnote 20).
22
The factor 2 appears due to the orientifold projection.
54
C
Abelian discrete gauge symmetries in Z2 × Z02
In this appendix we describe a different kind of discrete gauge symmetries that arise for
models of D-branes on the Z2 × Z02 orientifold. As shown in [7], massive U(1) D-brane
symmetries have Zk subgroups that survive as discrete gauge symmetries in the low energy
effective action. Even if they are Abelian and typically flavor blind, one should take them
into account in order to describe the full 4d discrete gauge group of a given D-brane
model. In the following we will apply the general discussion of [7] to the particular case of
type IIA models in the Z2 × Z02 orientifold. We refer the reader to [13] for a more detailed
discussion of these symmetries in the context of toroidal orientifolds.
Let us consider type IIA orientifold compactifications with D6-branes. Following [6],
we may take a linear combination of the D-brane U (1) symmetries and associate a threecycle Πα of the manifold to the resulting U (1)α , which will be a linear combination of the
three-cycles wrapped by the D6-branes of the model. A U (1)α is massless if
0
[Π−
α ] ≡ [Πα ] − [Πα ] = 0
(C.1)
That is, if Πα minus its orientifold image Π0α is trivial in homology. All the other U(1)’s
are massive, and will be broken by D2-brane instanton effects. Nevertheless, a Zk gauge
symmetry will remain if the intersection numbers of all possible D2-instantons with Π−
α
are a multiple of k. In practice that amounts to check that [7]
[Πα ] · ([Πβ ] + [Π0β ]) = 0 mod k
(C.2)
for any three-cycle Πβ of the compactification manifold. For the case of the Z2 × Z02
orientifold, this means that Πβ runs over all possible fractional three-cycles of the form
(A.3), that we can represent as
ΠFβ = (n1β , m1β ) (n2β , m2β ) (n3β , m3β )
(C.3)
and then we must remember for each choice (C.3) there are 8 different choices of discrete
positions (sets of fixed points where the three-cycles can go through) and 8 different
choices of discrete Wilson lines (choices for gij, a ). In terms of the action of the orientifold,
fractional three-cycles can be classified as follows:
55
Fully invariant three-cycles
These are the fractional three-cycles of the form:
(1, 0) (1, 0) (1, 0)
(1, 0) (−1, 0) (−1, 0)
(C.4)
(−1, 0) (1, 0) (−1, 0)
(−1, 0) (−1, 0) (1, 0)
that are fully invariant under the orientifold action. In this case the orientifold projection is such that a D2-brane instanton wrapping these cycles will develop a gauge group
projected down to O(1), and so we should not include Π0β in (C.2). For each of these four
fractional three-cycles we have 64 possibilities for their fractional positions and Wilson
lines. The computation of the intersection number between a fractional brane
Na ΠFa = Na (n1a , m1a ) (n2a , m2a ) (n3a , m3a )
(C.5)
and the three-cycles (C.4) is quite similar to the discussion of Appendix A of [16]. The
result is then that the intersection numbers are of the form
1
N
4 a
[m1a m2a m3a + Qa1 m1a + Qa2 m2a + Qa3 m3a ]
1
N
4 a
[m1a m2a m3a + Qa1 m1a − Qa2 m2a − Qa3 m3a ]
1
N
4 a
[m1a m2a m3a − Qa1 m1a + Qa2 m2a − Qa3 m3a ]
1
N
4 a
[m1a m2a m3a − Qa1 m1a − Qa2 m2a + Qa3 m3a ]
(C.6)
with Qia integer numbers that depend on the specific three-cycle ΠFa that the D-brane is
wrapping. More precisely,



1
if mja mka ≡ 1 mod 2



Qai =
2 and 0 if mja mka ≡ 0 mod 2 and mja + mka ≡ 1 mod 2




 4 and 0 if mj mk ≡ 0 mod 2 and mj + mk = 0 mod 2
a a
a
a
(C.7)
For instance, let us take the fractional three-cycle
ΠFa2 = (1, 0) (2, 1) (4, −1)
56
(C.8)
Then we have that the intersection numbers are given by (C.6) with
Qa22 = Qa33 = 0 or Qa22 = Qa33 = 2
(C.9)
in agreement with the observations above. The intersection numbers with the fully invariant three-cycles (C.4) are then
0, ±1
(C.10)
so that we already know that the g.c.d. of the intersection numbers is 1 and no discrete
gauge symmetry remains except the center of the gauge group SU (Na2 ).
Bulk invariant three-cycles
There are some other kind of fractional three-cycles whose bulk piece ΠB is invariant under
the action of the orientifold but that is not true for its fractional three-cycle content. One
example is given by
(1, 0) (0, 1) (0, −1)
ΩR
−→
(1, 0) (0, −1) (0, 1)
(C.11)
Hence, we need to consider invariant combinations [Πβ ]+[Π0β ] of these three-cycles, namely
[(1, 0) (0, 1) (0, −1)] + [(1, 0) (0, −1) (0, 1)]
[(−1, 0) (0, 1) (0, 1)] + [(−1, 0) (0, −1) (0, −1)]
[(0, 1) (1, 0) (0, −1)] + [(0, −1) (1, 0) (0, 1)]
(C.12)
[(0, 1) (−1, 0) (0, 1)] + [(0, −1) (−1, 0) (0, −1)]
[(0, −1) (0, 1) (1, 0)] + [(0, 1) (0, −1) (1, 0)]
[(0, 1) (0, 1) (−1, 0)] + [(0, −1) (0, −1) (−1, 0)]
The intersection numbers that we obtain with (C.5) are
1
N
2 a
[−m1a n2a n3a ± Qa1 m1a ]
1
N
2 a
[−n1a m2a n3a ± Qa2 m2a ]
1
N
2 a
[−n1a n2a m3a ± Qa3 m3a ]
(C.13)
where
Qai =






1
if nja nka ≡ 1 mod 2
2 and 0 if nja nka ≡ 0 mod 2 and nja + nka ≡ 1 mod 2




 4 and 0 if nj nk ≡ 0 mod 2 and nj + nk = 0 mod 2
a a
a
a
57
(C.14)
considering all the possibilities for the discrete positions and Wilson lines. For instance,
taking again the example (C.8) we have that
Qa22 = 0, 2 and Qa33 = 0, 2
(C.15)
and so the intersection numbers are 0, ±1, ±2.
Bulk anti-invariant three-cycles
Let us now consider the fractional three-cycles of the form
[(1, 0) (1, 0) (0, 1)]
ΩR
−→
[(1, 0) (1, 0) (0, −1)]
(C.16)
whose bulk piece is anti-invariant under the action of the orientifold. Hence the orientifold
invariant combination
[(0, 1) (1, 0) (1, 0)] + [(0, −1) (1, 0) (1, 0)]
(C.17)
just contains linear combinations of twisted cycles. The intersection number with (A.3)
is now
1 Na ±Qa2 m2a ± Qa3 m3a
2
(C.18)
where the rules for the values of Qia can be deduced as before. Considering other threecycles of this form we also obtain intersection numbers such as
1
N
2 a
[±Qa1 m2a ± Qa2 m3a ]
1
N
2 a
[±Qa1 m2a
±
(C.19)
Qa3 m3a ]
In the following we will assume that these above set of three-cycles generate all the
invariant homology classes of our toroidal orbifold.
The four-generation Pati-Salam example
As an example let us consider the four-generation Pati-Salam model of table 4 and let us
check whether there is any non-trivial Abelian discrete symmetry. The D6-brane stack a2
corresponds to two D6-branes wrapping (C.8), and so the Z2 symmetry that arises from
the fact that Na2 = 2 corresponds to the center of the gauge group SU (2).
58
In addition we have that the stack a1 corresponds to 4ΠFa1 with
ΠFa1 = (1, 0) (0, 1) (0, −1)
(C.20)
and so all the intersection numbers (C.6) vanish identically. If we now consider the
invariant combination (C.17) we see that Qa21 = Qa31 = 1, and so the intersection number
with ΠFa1 can be made equal to one for some choice of signs.
Finally, we consider the stack a3 corresponding to 2ΠFa3 with
ΠFa2 = (−3, 2) (−2, 1) (−4, 1)
(C.21)
Qa13 = 1 and Qa22 = Qa33 = 0 or Qa22 = Qa33 = 2
(C.22)
One can check that
and so the intersection numbers arising from (C.6) are
2 × (0, 1, 2)
(C.23)
where the factor of 4 comes from Na3 = 2 and it correspond to the Z2 within SU (2).
Hence, we conclude that in the Pati-Salam model at hand there is no non-trivial
discrete gauge symmetry which is a remnant of the massive U (1) symmetries, at least in
the visible sector of the model. As we already know, there will be nevertheless non-trivial
discrete flavor symmetries.
59
References
[1] For reviews see, e.g.,
G. Altarelli and F. Feruglio, “Discrete Flavor Symmetries and Models of Neutrino
Mixing,” Rev. Mod. Phys. 82, 2701 (2010) [arXiv:1002.0211 [hep-ph]].
H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, “NonAbelian Discrete Symmetries in Particle Physics,” Prog. Theor. Phys. Suppl. 183, 1
(2010) [arXiv:1003.3552 [hep-th]].
W. Grimus and P. O. Ludl, J. Phys. A 45, 233001 (2012) [arXiv:1110.6376 [hep-ph]].
S. Morisi and J. W. F. Valle, “Neutrino masses and mixing: a flavour symmetry
roadmap,” arXiv:1206.6678 [hep-ph].
S. F. King and C. Luhn, “Neutrino Mass and Mixing with Discrete Symmetry,” Rept.
Prog. Phys. 76, 056201 (2013) [arXiv:1301.1340 [hep-ph]].
[2] G. Altarelli, F. Feruglio and Y. Lin, “Tri-bimaximal neutrino mixing from orbifolding,” Nucl. Phys. B 775, 31 (2007) [hep-ph/0610165].
[3] T. Kobayashi, H. P. Nilles, F. Ploger, S. Raby and M. Ratz, “Stringy origin of nonAbelian discrete flavor symmetries,” Nucl. Phys. B 768, 135 (2007) [hep-ph/0611020].
H. P. Nilles, M. Ratz and P. K. S. Vaudrevange, “Origin of Family Symmetries,”
arXiv:1204.2206 [hep-ph].
[4] R. Kallosh, A. D. Linde, D. A. Linde and L. Susskind, “Gravity and global symmetries,” Phys. Rev. D 52, 912 (1995) [hep-th/9502069].
[5] T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and Gravity,”
Phys. Rev. D 83, 084019 (2011) [arXiv:1011.5120 [hep-th]].
[6] P. G. Cámara, L. E. Ibáñez and F. Marchesano, “RR photons,” JHEP 1109, 110
(2011) [arXiv:1106.0060 [hep-th]].
[7] M. Berasaluce-González, L. E. Ibáñez, P. Soler and A. M. Uranga, “Discrete gauge
symmetries in D-brane models,” JHEP 1112, 113 (2011) [arXiv:1106.4169 [hep-th]].
60
[8] M. Berasaluce-González, P. G. Cámara, F. Marchesano, D. Regalado and
A. M. Uranga, “Non-Abelian discrete gauge symmetries in 4d string models,” JHEP
1209, 059 (2012) [arXiv:1206.2383 [hep-th]].
[9] M. Berasaluce-González, P. G. Cámara, F. Marchesano and A. M. Uranga, “Zp
charged branes in flux compactifications,” JHEP 1304, 138 (2013) [arXiv:1211.5317
[hep-th]].
[10] M. Berasaluce-González, M. Montero, A. Retolaza and A. M. Uranga, “Discrete
gauge symmetries from (closed string) tachyon condensation,” arXiv:1305.6788 [hepth].
[11] L. E. Ibáñez, A. N. Schellekens and A. M. Uranga, “Discrete Gauge Symmetries in
Discrete MSSM-like Orientifolds,” Nucl. Phys. B 865, 509 (2012) [arXiv:1205.5364
[hep-th]].
[12] P. Anastasopoulos, M. Cvetic, R. Richter and P. K. S. Vaudrevange, “String Constraints on Discrete Symmetries in MSSM Type II Quivers,” JHEP 1303, 011 (2013)
[arXiv:1211.1017 [hep-th]].
[13] G. Honecker and W. Staessens, “To Tilt or Not To Tilt: Discrete Gauge Symmetries
in Global Intersecting D-Brane Models,” arXiv:1303.4415 [hep-th].
[14] For an overview of string phenomenology and model building see
L.E. Ibáñez and A.M. Uranga, String Theory and Particle Physics: An Introduction
to String Phenomenology, Cambridge University Press (2012).
[15] R. Blumenhagen, M. Cvetič, P. Langacker and G. Shiu, “Toward realistic intersecting
D-brane models,” Ann. Rev. Nucl. Part. Sci. 55, 71 (2005) [hep-th/0502005].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, “Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes,” Phys. Rept. 445, 1 (2007)
[hep-th/0610327].
F. Marchesano, “Progress in D-brane model building,” Fortsch. Phys. 55, 491 (2007)
[hep-th/0702094 [HEP-TH]].
61
[16] R. Blumenhagen, M. Cvetič, F. Marchesano and G. Shiu, “Chiral D-brane models
with frozen open string moduli,” JHEP 0503, 050 (2005) [hep-th/0502095].
[17] C. Angelantonj, I. Antoniadis, G. D’Appollonio, E. Dudas and A. Sagnotti, “Type
I vacua with brane supersymmetry breaking,” Nucl. Phys. B 572, 36 (2000) [hepth/9911081].
[18] E. Dudas and C. Timirgaziu, “Internal magnetic fields and supersymmetry in orientifolds,” Nucl. Phys. B 716, 65 (2005) [hep-th/0502085].
[19] H. Abe, T. Kobayashi and H. Ohki, “Magnetized orbifold models,” JHEP 0809, 043
(2008) [arXiv:0806.4748 [hep-th]];
[20] H. Abe, K. S. Choi, T. Kobayashi and H. Ohki, “Three generation magnetized orbifold
models,” Nucl. Phys. B 814, 265 (2009) [arXiv:0812.3534 [hep-th]].
[21] H. Abe, K. -S. Choi, T. Kobayashi and H. Ohki, “Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane Models,” Nucl. Phys. B 820, 317 (2009)
[arXiv:0904.2631 [hep-ph]].
[22] H. Abe, K. -S. Choi, T. Kobayashi, H. Ohki and , “Magnetic flux, Wilson line and
orbifold,” Phys. Rev. D 80, 126006 (2009) [arXiv:0907.5274 [hep-th]].
[23] H. Abe, K. -S. Choi, T. Kobayashi and H. Ohki, “Flavor structure from magnetic fluxes and non-Abelian Wilson lines,” Phys. Rev. D 81, 126003 (2010)
[arXiv:1001.1788 [hep-th]].
[24] D. Cremades, L. E. Ibáñez and F. Marchesano, “Computing Yukawa couplings from
magnetized extra dimensions,” JHEP 0405, 079 (2004) [arXiv:hep-th/0404229].
[25] S. Förste and I. Zavala, “Oddness from Rigidness,” JHEP 0807, 086 (2008)
[arXiv:0806.2328 [hep-th]].
[26] R. Blumenhagen, L. Görlich, B. Körs and D. Lüst, “Noncommutative compactifications of type I strings on tori with magnetic background flux,” JHEP 0010, 006
(2000) [hep-th/0007024].
62
[27] G. Aldazabal, S. Franco, L. E. Ibáñez, R. Rabadán and A. M. Uranga, “Intersecting
brane worlds,” JHEP 0102, 047 (2001) [hep-ph/0011132].
[28] G. Aldazabal, S. Franco, L. E. Ibáñez, R. Rabadán and A. M. Uranga, “D = 4 chiral
string compactifications from intersecting branes,” J. Math. Phys. 42, 3103 (2001)
[hep-th/0011073].
[29] R. Blumenhagen, B. Körs and D. Lüst, “Type I strings with F flux and B flux,”
JHEP 0102, 030 (2001) [hep-th/0012156].
[30] L. E. Ibáñez, F. Marchesano and R. Rabadán, “Getting just the standard model at
intersecting branes,” JHEP 0111, 002 (2001) [hep-th/0105155].
[31] M. Berkooz, M. R. Douglas and R. G. Leigh, “Branes intersecting at angles,” Nucl.
Phys. B 480, 265 (1996) [hep-th/9606139].
[32] M. Cvetič, G. Shiu and A. M. Uranga, “Chiral four-dimensional N=1 supersymmetric
type 2A orientifolds from intersecting D6 branes,” Nucl. Phys. B 615, 3 (2001) [hepth/0107166].
[33] F. G. Marchesano Buznego, “Intersecting D-brane models,” hep-th/0307252.
[34] D. Cremades, L. E. Ibáñez and F. Marchesano, “Yukawa couplings in intersecting
D-brane models,” JHEP 0307, 038 (2003) [arXiv:hep-th/0302105].
[35] M. G. Alford, F. Wilczek, “Aharonov-Bohm Interaction of Cosmic Strings with Matter,” Phys. Rev. Lett. 62 (1989) 1071;
[36] L. M. Krauss, F. Wilczek, “Discrete Gauge Symmetry in Continuum Theories,” Phys.
Rev. Lett. 62 (1989) 1221;
[37] M. G. Alford, J. March-Russell and F. Wilczek, “Discrete Quantum Hair On Black
Holes And The Nonabelian Aharonov-bohm Effect,” Nucl. Phys. B 337 (1990) 695.
[38] J. Preskill, L. M. Krauss, “Local Discrete Symmetry And Quantum Mechanical
Hair,” Nucl. Phys. B341 (1990) 50-100.
63
[39] M. G. Alford, K. Benson, S. R. Coleman, J. March-Russell and F. Wilczek, “The
Interactions And Excitations Of Nonabelian Vortices,” Phys. Rev. Lett. 64 (1990)
1632 [Erratum-ibid. 65 (1990) 668].
[40] M. G. Alford, S. R. Coleman and J. March-Russell, “Disentangling nonAbelian discrete quantum hair,” Nucl. Phys. B 351 (1991) 735.
[41] M. G. Alford and J. March-Russell, “Discrete gauge theories,” Int. J. Mod. Phys. B
5 (1991) 2641.
[42] M. G. Alford, K. -M. Lee, J. March-Russell and J. Preskill, “Quantum field theory
of nonAbelian strings and vortices,” Nucl. Phys. B 384 (1992) 251 [hep-th/9112038].
[43] C. P. Burgess, J. P. Conlon, L-Y. Hung, C. H. Kom, A. Maharana and F. Quevedo,
“Continuous Global Symmetries and Hyperweak Interactions in String Compactifications,” JHEP 0807, 073 (2008) [arXiv:0805.4037 [hep-th]].
[44] A. Maharana, “Symmetry Breaking Bulk Effects in Local D-brane Models,” JHEP
1206, 002 (2012) [arXiv:1111.3047 [hep-th]].
[45] R. Rabadán, “Branes at angles, torons, stability and supersymmetry,” Nucl. Phys. B
620, 152 (2002) [hep-th/0107036].
[46] C. Vafa, “Modular Invariance and Discrete Torsion on Orbifolds,” Nucl. Phys. B
273, 592 (1986).
[47] G. Honecker and T. Ott, “Getting just the supersymmetric standard model at intersecting branes on the Z(6) orientifold,” Phys. Rev. D 70, 126010 (2004) [Erratumibid. D 71, 069902 (2005)] [hep-th/0404055].
[48] F. Gmeiner and G. Honecker, “Mapping an Island in the Landscape,” JHEP 0709,
128 (2007) [arXiv:0708.2285 [hep-th]].
[49] S. Förste and G. Honecker, “Rigid D6-branes on T 6 /(Z2 xZ2M xΩR) with discrete
torsion,” JHEP 1101, 091 (2011) [arXiv:1010.6070 [hep-th]].
64
[50] G. Honecker, M. Ripka and W. Staessens, “The Importance of Being Rigid: D6Brane Model Building on T 6 /Z2 xZ60 with Discrete Torsion,” Nucl. Phys. B 868, 156
(2013) [arXiv:1209.3010 [hep-th]].
[51] R. Blumenhagen, V. Braun, B. Kors and D. Lust, “Orientifolds of K3 and Calabi-Yau
manifolds with intersecting D-branes,” JHEP 0207, 026 (2002) [hep-th/0206038].
[52] A. M. Uranga, “Local models for intersecting brane worlds,” JHEP 0212, 058 (2002)
[hep-th/0208014].
[53] E. Palti, “Model building with intersecting D6-branes on smooth Calabi-Yau manifolds,” JHEP 0904, 099 (2009) [arXiv:0902.3546 [hep-th]].
[54] A. Font, L. E. Ibáñez and F. Marchesano, “Coisotropic D8-branes and modelbuilding,” JHEP 0609, 080 (2006) [hep-th/0607219].
[55] F. Marchesano and G. Shiu, “MSSM vacua from flux compactifications,” Phys. Rev.
D 71, 011701 (2005) [arXiv:hep-th/0408059]. “Building MSSM flux vacua,” JHEP
0411, 041 (2004) [arXiv:hep-th/0409132].
[56] R. Blumenhagen, V. Braun, T. W. Grimm and T. Weigand, “GUTs in Type IIB
Orientifold Compactifications,” Nucl. Phys. B 815, 1 (2009) [arXiv:0811.2936 [hepth]].
[57] P. G. Cámara and F. Marchesano, “Open string wavefunctions in flux compactifications,” JHEP 0910, 017 (2009) [arXiv:0906.3033 [hep-th]]. “Physics from open string
wavefunctions,” PoS E PS-HEP2009, 390 (2009).
[58] S. A. Abel and M. D. Goodsell, “Realistic Yukawa couplings through instantons in
intersecting brane worlds,” JHEP 0710, 034 (2007) [arXiv:hep-th/0612110].
[59] R. Blumenhagen, M. Cvetič, D. Lüst, R. Richter and T. Weigand, “Non-perturbative
Yukawa Couplings from String Instantons,” Phys. Rev. Lett. 100, 061602 (2008)
[arXiv:0707.1871 [hep-th]].
[60] M. Berg, J. P. Conlon, D. Marsh and L. T. Witkowski, “Superpotential desequestering in string models,” JHEP 1302, 018 (2013) [arXiv:1207.1103 [hep-th]].
65